HB-Continuous Mappings in L-Topological Space

Najah A. AlSaedi

Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, Saudi Arabia,.

DOI: 10.4236/apm.2024.145019                

Abstract

In this paper, we introduce and study the notion of HB-closed sets in L-topological space. Then, HB-convergence theory for L-molecular nets and L-ideals is established in terms of HB-closedness. Finally, we give a new definition of fuzzy H-continuous [1] which is called HB-continuity on the basis of the notion of H-bounded L-subsets in L-topological space. Then we give characterizations and properties by making use of HB-converges theory of L-molecular nets and L-ideals.

Keywords

L-Topological Space, HB-Closed Set, H-Bounded Set, HB-Continuous Mappings, HB-Convergence, L-Molecular Nets, L-Ideals

1. Introduction

Continuity and its weaker forms constitute an important and intensely investigated area in the field of general topological spaces. In 1975 Long and Hamlett [2] introduced the notion of H-continuity and it has been further investigated by many authors including Noiri [3] . In 1993 Moony [4] studied the notion of H-bounded sets and some new characterizations and properties of H-bounded sets are examined. In 1995 Dang and Behers [1] extended the notion of H-continuity to fuzzy topology, and introduced the notion of fuzzy H-continuous functions using the fuzzy compactness given by Mukherjee and Sinha [5] . However, the fuzzy compactness has some shortcomings, such as the Tychonoff product theorem does not hold, and it contradicts some kinds of separation axioms. Hence, the notion of fuzzy H-continuous functions in [1] is unsatisfactory. In this paper, we first define the concept of HB-closed sets by means of the concept of almost N-boundedness (H-bounded L-subsets). Then by making use of HB-closed sets we introduce and study the HB-convergence theory of L-molecular nets and L-ideals. Finally, we give a new definition of fuzzy H-continuous [1] which calls HB-continuity on the basis of the notions of HB-closedness in L-topological space. In section 3, we introduce the concepts of HB-closure (HB-interior) operator and HB-closed (HB-open) sets in L-topological spaces and their various properties are given. And with the help of these notions we introduce and study the concept of HB-limit point of L-molecular nets and L-ideals. In section 4, we introduce and study the concept HB-continuous by means of HB-closed set and we present its properties and study the relationship between it and L-continuous, H-continuous mappings. Finally, in section 5, some new interesting characterizations of HB-continuous mappings by HB-limit points of L-molecular nets and L-ideals are established.

2. Preliminaries

This paper $L=L\left(\le ,\vee ,\wedge ,\text{'}\right)$ denotes a completely distributive lattice with the smallest element 0 and the largest element 1 ( $0\ne 1$ ) and with an order reversing involution on it. An $\alpha \in L$ is called a molecule of L if $\alpha \ne 0$ and $\alpha \le \nu \vee \gamma$ implies $\alpha \le \nu$ or $\alpha \le \gamma$ for all $\nu ,\gamma \in L$ . The set of all molecules of L is denoted by $M\left(L\right)$ . Let X be a nonempty set. $L^X$ denotes the family of all mappings from X to L. The elements of $L^X$ are called L-subsets on X. $L^X$ can be made into a lattice by inducing the order and involution from L. We denote the smallest element and the largest element of $L^X$ by $0_X$ and $1_X$ , respectively. If $\alpha \in L$ , then the constant mapping $\underset{\_}{\alpha }:X\to \left\{\alpha \right\}$ is L-subset [6] . An L-point (or molecule on $L^X$ ), denoted by $x_{\alpha }$ , $\alpha \in M\left(L\right)$ is a L-subset which

is defined by $x_{\alpha }\left(y\right)=\{\begin{array}{l}\alpha :x=y\\ 0:x\ne y\end{array}$ .

The family of all molecules $L^X$ is denoted by $M\left(L^X\right)$ [7] . For $\Psi \subset L^X$ , we define $2^{\left(\Psi \right)}$ by the set $\left\{\omega \subset \Psi :\omega \text{is}\text{finite}\text{subfamily}\text{of}\Psi \right\}$ . An L-topology on X is a subfamily $\tau$ of $L^X$ closed under arbitrary unions and finite intersections. The pair $\left(L^X,\tau \right)$ is called an L-topological space (or L-ts, for short) [8] . If $\left(L^X,\tau \right)$ is an L-ts, then for each $\eta \in L^X$ , $cl\left(\eta \right)$ , $\mathrm{int}\left(\eta \right)$ and ${\eta }^{\prime }$ will denote the closure, interior and complement of $\eta$ . A mapping $f:L^X\to L^Y$ is said to be an L-valued Zadeh function induced by a mapping $f:X\to Y$ , iff $f\left(\mu \right)\left(y\right)=\vee \left\{\mu \left(x\right):f\left(x\right)=y\right\}$ for every $\mu \in L^X$ and every $y\in Y$ [7] . An L-ts $\left(L^X,\tau \right)$ is called fully stratified if for each $\alpha \in L$ , $\underset{\_}{\alpha }\in \tau$ [9] . If $\left(L^X,\tau \right)$ is an L-ts, then the family of all crisp open sets in $\tau$ is denoted by $\left[\tau \right]$ i.e., $\left(X,\left[\tau \right]\right)$ is a crisp topological space [10] .

Definition 2.1 [11] : If $\left(L^X,\tau \right)$ is L-ts, then $\mu \in L^X$ is called regular open set iff $\mu =\mathrm{int}\left(cl\left(\mu \right)\right)$ . The family of all regular open sets is denoted by $RO\left(L^X,\tau \right)$ . The complement of the regular open set is called the regular closed set and satisfy $\mu =cl\left(\mathrm{int}\left(\mu \right)\right)$ . The family of all regular closed sets is denoted by $RC\left(L^X,\tau \right)$ .

Definition 2.2 [11] : The L-valued Zadeh mapping $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$

is called:

(i) Almost L-continuous iff $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ for each $\eta \in RC\left(L^Y,\Delta \right)$ .

(ii) Weakly L-continuous iff $f_L^{-1}\left(\eta \right)\le \mathrm{int}\left(f_L^{-1}\left(cl\left(\eta \right)\right)\right)$ for each $\eta \in \Delta$ .

Definition 2.3 [12] : Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-valued Zadeh mapping and $A\subseteq X$ , then ${f_L|}_A:L^A\to L^Y$ is defined as follows:

$\left({f_L|}_A\right)\left(\mu \right)=f\left(\mu \right)\wedge 1_A=f\left({\mu }^{\ast }\right)$ , for each $\mu \in L^A$ and call ${f_L|}_A$ the restriction of f on A. Where ${\mu }^{\ast }$ denote the extension of $\mu$ in $L^X$ , that is for each $x\in X$ ,

${\mu }^{\ast }\left(x\right)=\{\begin{array}{l}\mu \left(x\right):x\in A\\ 0:x\notin A\end{array}$

Definition 2.4 [13] : Let $\left(L^X,\tau \right)$ be an L-ts and $x_{\alpha }\in M\left(L^X\right)$ . Then:

(i) $\eta \in {\tau }^{\prime }$ is called a remote neighborhood (R-nbd, for short) of $x_{\alpha }$ if $x_{\alpha }\notin \eta$ . The set of all R-nbds of $x_{\alpha }$ is called remoted neighborhood system and

is denoted by $R_{x_{\alpha }}$ .

(ii) $\lambda \in L^X$ is called an $\ast$ -remoted neighborhood ( $R^{\ast }$ -nbd, for short ) of $x_{\alpha }$ if there exists $\mu \in R_{x_{\alpha }}$ such that $\lambda \le \mu$ . The set of all $R^{\ast }$ -nbds of $x_{\alpha }$ is

called $\ast$ -remoted neighborhood system and is denoted by $R_{x_{\alpha }}^{\ast }$ .

Definition 2.5 [14] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and $\alpha \in M\left(L\right)$ . Then $\Psi \subset {\tau }^{\prime }$ is called an:

(i) $\alpha$ -remoted neighborhood family of $\mu$ , briefly $\alpha$ -RF of $\mu$ , if for each

L-point $x_{\alpha }\in \mu$ there is $\lambda \in \Psi$ such that $\lambda \in R_{x_{\alpha }}$ .

(ii) $\bar{\alpha }$ -remoted neighborhood family of $\mu$ , briefly $\bar{\alpha }$ -RF of $\mu$ , if there exists $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ such that $\Psi$ is an $\gamma$ -RF of $\mu$ , where ${\beta }^{\ast }\left(\alpha \right)=\beta \left(\alpha \right)\cap M\left(L\right)$ , and $\beta \left(\alpha \right)$ denotes the union of all the minimal sets relative to $\alpha$ .

Definition 2.6 [11] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and $\alpha \in M\left(L\right)$ . Then $\Psi \subset {\tau }^{\prime }$ is called an:

(i) Almost $\alpha$ - $\ast$ -remoted neighborhood family of $\mu$ , (or briefly, almost $\alpha$ - $R^{\ast }F$ ) of $\mu$ , if for each L-point $x_{\alpha }\in \mu$ there is $\lambda \in \Psi$ such that

$\mathrm{int}\left(\lambda \right)\in R_{x_{\alpha }}^{\ast }$ .

(ii) Almost $\bar{\alpha }$ - $\ast$ -remoted neighborhood family of $\mu$ , (or briefly almost $\bar{\alpha }$ - $R^{\ast }F$ ) of $\mu$ , if there exists $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ such that $\Psi$ is an almost $\gamma$ - $R^{\ast }F$ of $\mu$ .

Definition 2.7 [15] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and $\alpha \in M\left(L\right)$ . Then $\Psi \subset RC\left(L^X,\tau \right)$ is called an $\alpha$ -regular closed remoted neighborhood family of $\mu$ , briefly $\alpha$ -RCRF of $\mu$ , if for each L-point $x_{\alpha }\in \mu$ there is $\lambda \in \Psi$ such

that $\lambda \in R_{x_{\alpha }}$ .

Definition 2.8 [16] : Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then $x_{\alpha }\in M\left(L^X\right)$ is called $\theta$ -adherent point of $\mu$ and write $x_{\alpha }\in \theta .cl\left(\mu \right)$ iff $\mu \cancel{\le }\mathrm{int}\left(\lambda \right)$ for

each $\lambda \in R_{x_{\alpha }}$ . If $\mu =\theta .cl\left(\mu \right)$ , then $\mu$ is called $\theta$ -closed L-subset. The family

of all $\theta$ -closed L-subset of X is denoted by $\theta C\left(L^X,\tau \right)$ and its complement is called the family of all $\theta$ -open L-subset and denoted by $\theta O\left(L^X,\tau \right)$ .

Definition 2.9 [11] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ . Then $\mu$ is called almost N-compact (or H-compact) set in $\left(L^X,\tau \right)$ if for each $\alpha \in M\left(L\right)$ and every $\alpha$ -RF $\Psi$ of $\mu$ there is ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ .

If $1_X$ is H-compact set, then $\left(L^X,\tau \right)$ is called H-compact space.

Theorem 2.10 [11] : Suppose that $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is an L-almost continuous and $\mu \in L^X$ is an H-compact L-subset in $\left(L^X,\tau \right)$ , then $f_L\left(\mu \right)$ is an H-compact L-subset in $\left(L^Y,\Delta \right)$ .

Definition 2.11 [17] : An L-ts $\left(L^X,\tau \right)$ is said to be:

(i) $L{T}_1$ -space iff for any $x_{\alpha },y_{\gamma }\in M\left(L^X\right)$ , $x\ne y$ there is $\lambda \in R_{x_{\alpha }}$ such that $y_{\gamma }\in \lambda$ .

(ii) $L{T}_2$ -space iff for any $x_{\alpha },y_{\gamma }\in M\left(L^X\right)$ , $x\ne y$ there is $\lambda \in R_{x_{\alpha }}$ , $\eta \in R_{y_{\gamma }}$ such that $\lambda \vee \eta =1_X$ .

(iii) $L{T}_{2\frac{1}{2}}$ -space iff for any $x_{\alpha },y_{\gamma }\in M\left(L^X\right)$ , $x\ne y$ there is $\lambda \in R_{x_{\alpha }}$ , $\eta \in R_{y_{\gamma }}$ such that $\mathrm{int}\left(\lambda \right)\vee \mathrm{int}\left(\eta \right)=1_X$ .

(iv) $L{R}_2$ -space (regular space) iff for all $\alpha \in M\left(L\right)$ , $x\in X$ and for each $\lambda \in R_{x_{\alpha }}$ there is $\eta \in R_{x_{\alpha }}$ , $\rho \in {\tau }^{\prime }$ such that $\eta \vee \rho =1_X$ and $\lambda \wedge \rho =0_X$ .

(v) $L{T}_3$ -space iff it is $L{R}_2$ -space and $L{T}_1$ -space.

Theorem 2.12 [14] : Let $\left(L^X,\tau \right)$ be an L-ts and every H-compact set in fully

stratified and $L{T}_{2\frac{1}{2}}$ -space, then it is $\theta$ -closed L-subset.

Theorem 2.13 [11] : An L-ts $\left(L^X,\tau \right)$ is $L{R}_2$ -space iff for any $\mu \in L^X$ , $cl\left(\mu \right)=\theta .cl\left(\mu \right)$ .

Proof. Let $\left(L^X,\tau \right)$ be an $L{R}_2$ -space. For any $\mu \in L^X$ it is always true that

$cl\left(\mu \right)\le \theta .cl\left(\mu \right)$ . Now, let $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\notin cl\left(\mu \right)$ and let $\lambda \in R_{x_{\alpha }}$ ,

since $\left(L^X,\tau \right)$ is $L{R}_2$ -space, there is $\eta \in R_{x_{\alpha }}$ such that $\lambda \le \mathrm{int}\left(\eta \right)$ . Now $x_{\alpha }\notin cl\left(\mu \right)$ implies that $\mu \le \lambda$ for each $\lambda \in R_{x_{\alpha }}$ which implies that $\mu \le \mathrm{int}\left(\eta \right)$ which implies that $x_{\alpha }\notin \theta .cl\left(\mu \right)$ . Thus $\theta .cl\left(\mu \right)\le cl\left(\mu \right)$ . Hence $cl\left(\mu \right)=\theta .cl\left(\mu \right)$ . Conversely, let $x_{\alpha }\in M\left(L^X\right)$ and $\lambda \in R_{x_{\alpha }}$ . Then $cl\left(\lambda \right)\in R_{x_{\alpha }}$ and so $x_{\alpha }\notin cl\left(\lambda \right)=\theta .cl\left(\lambda \right)$ . Hence there is $\eta \in R_{x_{\alpha }}$ such that $\lambda \le \mathrm{int}\left(\eta \right)$ .

Thus $\left(L^X,\tau \right)$ is $L{R}_2$ -space.

Corollary 2.14 [11] : If $\left(L^X,\tau \right)$ is $L{R}_2$ -space, then closed L-subset is $\theta$ -closed L-subset and hence $\theta .cl\left(\mu \right)$ is $\theta$ -closed for any $\mu \in L^X$ .

Definition 2.15 [13] : Let $\left(D,\le \right)$ be a directed set. Then the mapping $S:D\to L^X$ and denoted by $S=\left\{{\mu }_n:n\in D\right\}$ is called a net of L-subsets in X. Specially, the mapping $S:D\to M\left(L^X\right)$ is said to be a molecular net in $L^X$ . If $\mu \in L^X$ and for each $n\in D$ , $S\in \mu$ then S is called a net in $\mu$ .

Definition 2.16 [13] : Let $\left(L^X,\tau \right)$ be an L-ts and $S=\left\{S\left(n\right):n\in D\right\}$ be a molecular net in $L^X$ . S is called a molecular $\alpha$ -net ( $\alpha \in M\left(L\right)$ ), if for each $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ there exists $n\in D$ such that $\vee \left(S\left(m\right)\right)\ge \gamma$ whenever $m\ge n$ , where $\vee \left(S\left(m\right)\right)$ is the height of the molecular $S\left(m\right)$ .

Definition 2.17 [13] : Let $S=\left\{S\left(n\right):n\in D\right\}$ and $T=\left\{T\left(m\right):m\in E\right\}$ be a be molecular nets in $\left(L^X,\tau \right)$ . Then T is said to be a molecular subnet of S if there is a mapping $f:E\to D$ that satisfies the following conditions:

(i) $T=S\circ f$

(ii) For each $n\in D$ there is $m\in E$ such that $f\left(l\right)\ge n$ for each $l\in E$ , $l\ge m$ .

Definition 2.18 [7] : Let $\left(L^X,\tau \right)$ be an L-ts and S be a molecular net in $\left(L^X,\tau \right)$ . Then $x_{\alpha }\in M\left(L^X\right)$ is called:

(i) a $\theta$ -limit point of S, (or S $\theta$ -converges to $x_{\alpha }$ ) in symbols $S\stackrel{\theta }{\to }{x}_{\alpha }$ if

for each $\mu \in R_{x_{\alpha }}$ there is a $n\in D$ such for each $m\in D$ and $m\ge n$ we have

$S\left(m\right)\notin \mathrm{int}\left(\mu \right)$ . The union of all $\theta$ -limit points of S are denoted by $\theta .\lim \left(S\right)$ .

(ii) a $\theta$ -cluster ( $\theta$ -adherent) point of S, in symbols $S\stackrel{\theta }{\propto }{x}_{\alpha }$ if for each $\mu \in R_{x_{\alpha }}$ and for each $n\in D$ there is a $m\in D$ such that $m\ge n$ and

$S\left(m\right)\notin \mathrm{int}\left(\mu \right)$ . The union of all $\theta$ -cluster points of S is denoted by $\theta .adh\left(S\right)$ .

Theorem 2.19 [13] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and $x_{\alpha }\in M\left(L^X\right)$ . Then $x_{\alpha }\in \theta \mathrm{..}cl\left(\mu \right)$ iff there exists a molecular net S in $\mu$ such that S is $\theta$ -converges to $x_{\alpha }$ .

Theorem 2.20 [15] : Assume that $S=\left\{S\left(n\right):n\in D\right\}$ is a molecular net in an

L-ts $\left(L^X,\tau \right)$ and $x_{\alpha }\in M\left(L^X\right)$ . Then $S\stackrel{\theta }{\propto }{x}_{\alpha }$ iff there exists a subnet T of S

such that $T\stackrel{\theta }{\to }{x}_{\alpha }$ .

Theorem 2.21 [14] : Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then $\mu$ is H-compact set iff each $\alpha$ -net S contained in $\mu$ has a $\theta$ -cluster point in $\mu$ with height $\alpha$ for any $\alpha \in M\left(L\right)$ .

Definition 2.22 [18] : The nonempty family $I\subset L^X$ is called an ideal if the following conditions are satisfied, for each ${\mu }_1,{\mu }_2\in L^X$

(i) $1_X\notin I$

(ii) If ${\mu }_1\le {\mu }_2$ and ${\mu }_2\in I$ , then ${\mu }_1\in I$ .

(iii) If ${\mu }_1,{\mu }_2\in I$ , then ${\mu }_1\vee {\mu }_2\in I$ .

Theorem 2.23 [19] : Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and $x_{\alpha }\in M\left(L^X\right)$ . Then $x_{\alpha }\in \theta \mathrm{..}cl\left(\mu \right)$ iff there exists an ideal I in $L^X$ such that I is $\theta$ -converges to $x_{\alpha }$ and $\mu \notin I$ .

Definition 2.24 [20] : An L-mapping $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is called H-continuous if $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ for each $\eta \in L^Y$ is closed and almost N-compact.

3. H-Closure and H-Interior Operators in L-Topological Space

In this section, we introduce the concepts of H-Closure operator and H-interior operator by using an almost N-bounded (or H-bounded) set and discuss their properties.

Definition 3.1: Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ . Then $\mu$ is called almost N-bounded (or H-bounded) set in $\left(L^X,\tau \right)$ if for each $\alpha \in M\left(L\right)$ and every $\alpha$ -RF $\Psi$ of $1_X$ , there is ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ .

If $1_X$ is H-bounded set, then $\left(L^X,\tau \right)$ is called H-bounded space.

Theorem 3.2: Suppose that $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is an L-almost continuous and $\mu \in L^X$ is an H-bounded L-subset in $\left(L^X,\tau \right)$ , then $f_L\left(\mu \right)$ is an H-bounded L-subset in $\left(L^Y,\Delta \right)$ .

Proof. Let $\mu$ be an H-bounded in $L^X$ and let $\Psi \subseteq {\Delta }^{\prime }$ be an $\alpha$ -RF of $1_Y$

( $\alpha \in M\left(L\right)$ ), then $\left\{cl\left(\mathrm{int}\left(\lambda \right)\right):\lambda \in \Psi \right\}\subset RC\left(L^Y,\Delta \right)$ is an $\alpha$ -RCRF of $1_Y$ . We now will show that $Q=\left\{f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right):\lambda \in \Psi \right\}$ is an $\alpha$ -RF of $1_X$ . In fact,

since $f_L$ is an L-almost continuous and $cl\left(\mathrm{int}\left(\lambda \right)\right)\in RC\left(L^Y,\Delta \right)$ then

$f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right)\in {\tau }^{\prime }$ . According to the definition, $\Psi$ there exists $\lambda \in \Psi$

such that $cl\left(\mathrm{int}\left(\lambda \right)\right)\in R_{f_L\left(x_{\alpha }\right)}$ , i.e., $f_L\left(x_{\alpha }\right)\notin cl\left(\mathrm{int}\left(\lambda \right)\right)$ hence

$x_{\alpha }\notin f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right)$ for every $x\in X$ . This means that Q is an $\alpha$ -RF of $1_X$ . Since $\mu$ is an H-bounded set, there exists ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that

$\left\{f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right):\lambda \in {\Psi }_{\circ }\right\}\in 2^{\left(\Psi \right)}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ . Thus for some $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ and for each $x_{\gamma }\in \mu$ there exists $\lambda \in {\Psi }_{\circ }$ such that

$\mathrm{int}\left(f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right)\right)\in R_{x_{\gamma }}^{\ast }$ . Since $f_L$ is an L-almost continuous then it is L-weakly continuous and since $\mathrm{int}\left(\lambda \right)\in \Delta$ then

$f_L^{-1}\left(\mathrm{int}\left(\lambda \right)\right)\le \mathrm{int}\left(f_L^{-1}\left(cl\left(\mathrm{int}\left(\lambda \right)\right)\right)\right)$ and so $x_{\alpha }\notin f_L^{-1}\left(\mathrm{int}\left(\lambda \right)\right)$ . Consequently, there exists $x_{\gamma }\in \mu$ and $\lambda \in {\Psi }_{\circ }$ satisfying $\mathrm{int}\left(\lambda \right)\in R_{f_L\left(x_{\gamma }\right)}^{\ast }$ and $y_{\gamma }=f_L(\; x\; \gamma \; )$

for each $y_{\gamma }\in f_L\left(\mu \right)$ . Thus, ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $f_L\left(\mu \right)$ . By Definition 3.1, we have $f_L\left(\mu \right)$ an H-bounded L-subset in $\left(L^Y,\Delta \right)$ .

Theorem 3.3: Let $\left(L^X,\tau \right)$ be an L-ts and let $\mu \in L^X$ . Then the following statements are true:

(i) If $\mu$ is H-compact set, then $\mu$ is H-bounded set.

(ii) If $\mu$ is H-bounded set and $\eta \le \mu$ , then $\eta$ is H-bounded set.

(iii) If $\mu$ is H-compact set and $\eta \le \mu$ , then $\eta$ is H-bounded set.

Proof. (i) Let $\mu$ be an H-compact set and let $\Psi =\left\{{\rho }_i:i\in I\right\}\subset {\tau }^{\prime }$ be an $\alpha$ -RF of $1_X$ and so $\Psi$ is $\alpha$ -RF of $\mu$ . Since $\mu$ is H-compact set, then there exists ${\Psi }_{\circ }=\left\{{\rho }_i:i=1,2,\cdots ,m\right\}\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ . Thus $\mu$ is H-bounded set.

(ii) Let $\mu$ be an H-bounded set and $\eta \le \mu$ . let $\Psi =\left\{{\rho }_i:i\in I\right\}\subset {\tau }^{\prime }$ be an $\alpha$ -RF of $1_X$ . Since $\mu$ is H-bounded set, then there exists

${\Psi }_{\circ }=\left\{{\rho }_i:i=1,2,\cdots ,m\right\}\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ , thus there exists $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ such that ${\Psi }_{\circ }$ is an almost $\gamma$ - $R^{\ast }F$ of $\mu$ . Hence

$\forall x_{\gamma }\in \mu$ , $\exists \lambda \in {\Psi }_{\circ }$ such that $\mathrm{int}\left(\lambda \right)\in R_{x_{\gamma }}^{\ast }$ . Since $\eta \le \mu$ , then $\forall x_{\gamma }\in \eta \le \mu$ ,

$\exists \lambda \in {\Psi }_{\circ }$ such that $\mathrm{int}\left(\lambda \right)\in R_{x_{\gamma }}^{\ast }$ . Hence ${\Psi }_{\circ }$ is an almost $\gamma$ - $R^{\ast }F$ of $\eta$ and

so ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\eta$ . Thus $\eta$ is H-bounded set.

(iii) Let $\mu$ be an H-compact set and $\eta \le \mu$ . let $\Psi \subset {\tau }^{\prime }$ be an $\alpha$ -RF of $1_X$ and so $\alpha$ -RF of $\mu$ . Since $\mu$ is H-compact set, then there exists ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ , since $\eta \le \mu$ , then ${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\eta$ . Thus $\eta$ is H-bounded set.

Theorem 3.4: Let $\left(L^X,\tau \right)$ be an L-ts, $\alpha \in M\left(L\right)$ and $\mu \in L^X$ . Then $\mu$ is H-bounded iff for each molecular $\alpha$-net S contained in $\mu$ has $\theta$ -cluster point in $1_X$ with height $\alpha$ .

Proof. Let $\mu$ be an H-bounded set and $S=\left\{S\left(n\right):n\in D\right\}$ be an molecular $\alpha$ -net in $\mu$ . If S does not have any $\theta$ -cluster point in $1_X$ with height $\alpha$ . Then for all $x_{\alpha }\in M\left(L^X\right)$ , $x_{\alpha }$ is not $\theta$ -cluster point of S and so there exists

${\lambda }_x\in R_{x_{\alpha }}$ and $n_x\in D$ such that $S\left(n\right)\in \mathrm{int}\left({\lambda }_x\right)$ for every $n\in D$ and $n\ge n_x$ .

Put $\Psi =\left\{{\lambda }_x:x\in X\text{and}\alpha \in M\left(L\right)\right\}$ , then $\Psi$ is an $\alpha$ -RF of $1_X$ . According to

the hypothesis, $\Psi$ has a finite family ${\Psi }_{\circ }=\left\{{\lambda }_{x^i}:i=1,2,\cdots ,k\right\}\in 2^{\left(\Psi \right)}$ such that

${\Psi }_{\circ }$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ , that is for some $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ and each

$y_{\gamma }\in \mu$ there exists ${\lambda }_{x_i}\in {\Psi }_{\circ }$ ( $i\le k$ ) such that $\mathrm{int}\left({\lambda }_{x_i}\right)\in R_{y_{\gamma }}^{\ast }$ . Put $\lambda =\underset{i=1}{\overset{k}{\wedge }}{\lambda }_{x_i}$ , for each $y_{\gamma }\in \mu$ , we have $\underset{i=1}{\overset{k}{\wedge }}\mathrm{int}\left({\lambda }_{x_i}\right)=\mathrm{int}\left(\underset{i=1}{\overset{k}{\wedge }}{\lambda }_{x_i}\right)=\mathrm{int}\left(\lambda \right)$ , thus $\mathrm{int}\left(\lambda \right)\in R_{y_{\gamma }}^{\ast }$ . Since D is a directed set, then there is $n_{\circ }\in D$ such that $n_{\circ }\ge n_{x_i}$ , $i=1,2,\cdots ,k$ and $S\left(n\right)\in \mathrm{int}\left({\lambda }_{x_i}\right)$ , $i=1,2,\cdots ,k$ whenever $n\ge n_{\circ }$ and so $S\left(n\right)\in \mathrm{int}\left(\lambda \right)$ .

This shows that for each $y_{\gamma }\in \mu$ , $\vee \left(S\left(n\right)\right)\cancel{\ge }\gamma$ whenever $n\ge n_{\circ }$ . This contradicts the hypothesis that S is a molecular $\alpha$ -net. Therefore, S has at least a $\theta$ -cluster point in $1_X$ with height $\alpha$ .

Conversely, assume that each molecular $\alpha$ -net S contained in $\mu$ has an $\theta$ -cluster point in $1_X$ with height $\alpha$ and $\Psi$ is an $\alpha$ -RF of $1_X$ . If for each ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is not almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ , that is, for each $\gamma \in {\beta }^{\ast }\left(\alpha \right)$ there exists $\left(\gamma ,{\Psi }_{\circ }\right)\in {\beta }^{\ast }\left(\alpha \right)\times 2^{\left(\Psi \right)}$ there exists molecule

$x_{\left(\gamma ,{\Psi }_{\circ }\right)}\in \mu$ such that for each $\lambda \in {\Psi }_{\circ }$ , $\mathrm{int}\left(\lambda \right)\notin R_{x_{\left(\gamma ,{\Psi }_{\circ }\right)}}$ . Put $D={\beta }^{\ast }\left(\alpha \right)\times 2^{\left(\Psi \right)}$ and defined the order as follows: $\left({\gamma }_1,{\Psi }_{\circ }^1\right)\ge \left({\gamma }_2,{\Psi }_{\circ }^2\right)$ iff ${\gamma }_1\ge {\gamma }_2$ and ${\Psi }_{\circ }^1\supset {\Psi }_{\circ }^2$ . Then $S=\left\{S_{\left(\gamma ,{\Psi }_{\circ }\right)}=x_{\left(\gamma ,{\Psi }_{\circ }\right)}\in \mu :\left(\gamma ,{\Psi }_{\circ }\right)\in D\right\}$ is an molecular $\alpha$ -net in $\mu$ . Since $\Psi$ is an $\alpha$ -RF of $1_X$ , then there exists $\rho \in \Psi$ such that $\rho \in R_{y_{\alpha }}$ and hence $\mathrm{int}\left(\rho \right)\in R_{x_{\alpha }}^{\ast }$ . Because $\left\{\rho \right\}\in 2^{\left(\Psi \right)}$ . We take any ${\gamma }_1\in {\beta }^{\ast }\left(\alpha \right)$ , $x_{\left(\gamma ,{\Psi }_{\circ }\right)}\in \mathrm{int}\left(\rho \right)$ whenever $\left(\gamma ,{\Psi }_{\circ }\right)\ge \left({\gamma }_1,\rho \right)$ . Therefore $S_{\left(\gamma ,{\Psi }_{\circ }\right)}\in \mathrm{int}\left(\rho \right)$ , which

contradicts to the hypothesis. Therefore there exists ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ such that ${\Psi }_{\circ }$ is almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ and hence $\mu$ is H-bounded.

Theorem 3.5: If $\left(L^X,\tau \right)$ fully stratified and $L{T}_{2\frac{1}{2}}$ -space, then $\mu \in L^X$ is H-compact set iff $\mu$ is $\theta$ -closed and H-bounded set.

Proof. If $\mu \in L^X$ is H-compact set, then by Theorem 2.12 we have $\mu$ is $\theta$ -closed and by Theorem 3.3 (i) we have $\mu$ is H-bounded. Conversely, let $\mu$ be an $\theta$ -closed and H-bounded set and let S be an $\alpha$ -net in $\mu$ . Since $\mu$ is H-bounded, then by Theorem 3.4 we have S has $\theta$ -cluster point, say $x_{\alpha }$ in $1_X$ with height $\alpha$ . By Theorem 2.20, then there is a subnet T of S such that T $\theta$ -converges to $x_{\alpha }$ and so $x_{\alpha }\in \theta .cl\left(\mu \right)$ by Theorem 2.19. Since $\mu$ is $\theta$ -closed, then $\mu =\theta .cl\left(\mu \right)$ and so $x_{\alpha }\in \mu$ , then by Theorem 2.21 we have $\mu$ is H-compact set.

Theorem 3.6: If $\left(L^X,\tau \right)$ is $L{R}_2$ -space, then $\mu \in L^X$ is H-bounded set iff $\theta .cl\left(\mu \right)$ is H-bounded set.

Proof. If $\theta .cl\left(\mu \right)$ is H-bounded set, then $\mu$ is H-bounded set by Theorem

3.3 (ii). Conversely, suppose that $\mu$ is H-bounded and $\Psi =\left\{{\eta }_{x_j}:j\in J\right\}$ is an $\alpha$ -RF of $1_X$ . Then for each $x\in X$ there is ${\eta }_{x_j}\in \Psi$ such that ${\eta }_{x_j}\in R_{x_{\alpha }}$ . Since $\left(L^X,\tau \right)$ is $L{R}_2$ -space, then there is $\lambda \in R_{x_{\alpha }}$ there is ${\lambda }_{x_j}\in R_{x_{\alpha }}$ and there is ${\rho }_{x_j}\in {\tau }^{\prime }$ such that ${\lambda }_{x_j}\vee {\rho }_{x_j}=1_X$ and. ${\rho }_{x_j}\wedge {\eta }_{x_j}=0_X$ . Then the family $\left\{{\lambda }_{x_j}:x_{\alpha }\in M\left(L^X\right)\right\}$ is an $\alpha$ -RF of $1_X$ . Since $\mu$ is H-bounded, then exists finite subset $J_{\circ }$ of J such that $\left\{{\lambda }_{x_j}:j\in J_{\circ }\right\}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\mu$ . Since ${\lambda }_{x_j}\vee {\rho }_{x_j}=1_X$ , $x_{\alpha }\notin {\lambda }_{x_j}$ , then $x_{\alpha }\in {\rho }_{x_j}$ . Since ${\rho }_{x_j}\wedge {\eta }_{x_j}=0_X$ , then $\left\{{\eta }_{x_j}:j\in J_{\circ }\right\}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of ${\rho }_{x_j}$ . Therefore $\mu \le {\rho }_{x_j}$ for $J\in J_{\circ }$ . Since ${\rho }_{x_j}\in {\tau }^{\prime }$ , and $\left(L^X,\tau \right)$ is $L{R}_2$ -space, then by Theorem 2.13, we have $cl\left({\rho }_{x_j}\right)=\theta .cl\left({\rho }_{x_j}\right)$ and so $\left\{{\eta }_{x_j}:j\in J_{\circ }\right\}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of $\theta .cl\left({\rho }_{x_j}\right)$ and since $\theta .cl\left(\mu \right)\le \theta .cl\left({\rho }_{x_j}\right)$ , then $\left\{{\eta }_{x_j}:j\in J_{\circ }\right\}$ is an almost $\bar{\alpha }$ - $R^{\ast }F$ of

$\theta .cl\left(\mu \right)$ . Hence $\theta .cl\left(\mu \right)$ is H-bounded set.

Theorem 3.7: If $\left(L^X,\tau \right)$ is $L{T}_3$ -space, then $\mu \in L^X$ is H-bounded set iff $\mu$ is L-subset of H-compact set.

Proof. If $\mu$ is H-bounded, then by Theorem 3.6 and corollary 2.14, we have $\theta .cl\left(\mu \right)$ is $\theta$ -closed and H-bounded set, hence by Theorem 3.5, we have $\theta .cl\left(\mu \right)$ is H-compact set. Conversely, If $\mu$ is L-subset of H-compact set, then by Theorem 3.3 (iii), we have $\mu$ is H-bounded set.

Definition 3.8: Let $\left(L^X,\tau \right)$ be an L-ts and $x_{\alpha }\in M\left(L^X\right)$ . If $\mu \in L^X$ is closed and H-bounded set, then $\mu$ is called HB-remoted neighborhood of $x_{\alpha }$

(HBR-nbd, for short) of $x_{\alpha }$ if $x_{\alpha }\notin \mu$ . The set of all HBR-nbds of $x_{\alpha }$ is denoted by $HB{R}_{x_{\alpha }}$

We note that $HB{R}_{x_{\alpha }}\subseteq R_{x_{\alpha }}$ , $\forall x_{\alpha }\in M(\; L\; X\; )$

The following example shows that the converse is not true in general

Example 3.9: Let $X=\left\{x\right\}$ , $L=\left[0,1\right]$ , and let $\tau =\left\{0_X,x_3,x_{.7},1_X\right\}$ . Then $\left(L^X,\tau \right)$ is L-ts. We have $R_{x_1}=\left\{0_X,x_{.3},x_{.7}\right\}$ . Now, we show that $x_{.7}\in L^X$ is not H-bounded set.

Let $\Psi =\left\{x_{.7},1_X\right\}\subseteq {\tau }^{\prime }$ , then $\Psi$ is .8-RF of $1_X$ . But for each

$\gamma \in {\beta }^{\ast }\left(.8\right)=\left(0,2\right]$ , any finite subfamily ${\Psi }_{\circ }\in 2^{\left(\Psi \right)}$ is not almost $\gamma$ - $R^{\ast }F$ of $x_{.7}$ . Thus ${\Psi }_{\circ }$ is not almost $\overline{.8}$ - $R^{\ast }F$ of $x_{.7}$ . Thus $x_{.7}$ is not H-bounded set

and so $x_{.7}\notin HB{R}_{x_{\alpha }}$ . Hence $R_{x_{.7}}\cancel{\subseteq }HB{R}_{x_{.7}}$ .

Definition 3.10: Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then $x_{\alpha }\in M\left(L^X\right)$ is called an H-bounded adherent point of $\mu$ and write $x_{\alpha }\in HB.cl\left(\mu \right)$ iff

$\mu \cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ . If $\mu =HB.cl\left(\mu \right)$ , then $\mu$ is called HB-closed

L-subset. The family of all HB-closed L-subsets is denoted by $HBC\left(L^X,\tau \right)$ and its complement is called the family of all HB-open L-subsets and denoted by $HBO\left(L^X,\tau \right)$ .

Theorem 3.11: Let $\left(L^X,\tau \right)$ be an L-ts and let $\mu \in L^X$ . Then the following statements are true:

(i) $\mu \le cl\left(\mu \right)\le HB.cl\left(\mu \right)$ .

(ii) If $\eta \in L^X$ and $\mu \le \eta$ then $HB.cl\left(\mu \right)\le HB.cl\left(\eta \right)$ .

(iii) $HB.cl\left(HB.cl\left(\mu \right)\right)=HB.cl\left(\mu \right)$ .

(iv) $HB.cl\left(\mu \right)=\wedge \left\{\eta \in L^X:\eta \in HBC.\left(L^X,\tau \right),\mu \le \eta \right\}$.

Proof. (i) Let $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\notin HB.cl\left(\mu \right)$ , then there exists

$\lambda \in HB{R}_{x_{\alpha }}$ such that $\mu \le \lambda$ . Since $HB{R}_{x_{\alpha }}\subseteq R_{x_{\alpha }}$ and so $\lambda \in R_{x_{\alpha }}$ and hence

$x_{\alpha }\notin cl\left(\mu \right)$ . Thus $cl\left(\mu \right)\le HB.cl\left(\mu \right)$ .

(ii) Let $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\notin HB.cl\left(\eta \right)$ , then there exists $\lambda \in HB{R}_{x_{\alpha }}$

such that $\eta \le \lambda$ . Since $\mu \le \eta$ , then $\mu \le \lambda$ and so $x_{\alpha }\notin HB.cl\left(\mu \right)$ . Thus $HBcl\left(\mu \right)\le HB.cl\left(\eta \right)$ .

(iii) Suppose $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in HB.cl\left(HB.cl\left(\mu \right)\right)$ . According to

Definition 3.10, we have $HB.cl\left(\mu \right)\cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ . Hence, there exists $y_{\gamma }\in M\left(L^X\right)$ such that $y_{\gamma }\in HB.cl\left(\mu \right)$ with $y_{\gamma }\notin \lambda$ and so $\mu \cancel{\le }\lambda$ , that is,

$x_{\alpha }\in HB.cl\left(\mu \right)$ . This shows that $HB.cl\left(HB.cl\left(\mu \right)\right)\le HB.cl\left(\mu \right)$ . On the other hand, $\mu \le HB.cl\left(\mu \right)$ follows from (i) and so $HB.cl\left(\mu \right)\le HB.cl\left(HB.cl\left(\mu \right)\right)$ . Therefore, $HB.cl\left(HB.cl\left(\mu \right)\right)=HB.cl\left(\mu \right)$ .

(iv) On account of (i) and (iii). $HB.cl\left(\mu \right)$ is an HB-closed set containing $\mu$ ,

and so $HB.cl\left(\mu \right)\ge \wedge \left\{\eta \in L^X:\eta \in HBC.\left(L^X,\tau \right),\mu \le \eta \right\}$ . Conversely, in case

$x_{\alpha }\in M\left(L^X\right)$ sand $x_{\alpha }\in HB.cl\left(\mu \right)$ , then $\mu \cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ . Hence, if

$\eta$ is an HB-closed set containing $\mu$ , then $\eta \cancel{\le }\lambda$ , and then $x_{\alpha }\in HB.cl\left(\eta \right)=\eta$ .

This implies that $HB.cl\left(\mu \right)\le \wedge \left\{\eta \in L^X:\eta \in HBC.\left(L^X,\tau \right),\mu \le \eta \right\}$ . Hence

$HB.cl\left(\mu \right)=\wedge \left\{\eta \in L^X:\eta \in HBC.\left(L^X,\tau \right),\mu \le \eta \right\}$

From Theorem 3.11, one can see that every HB-closed L-subset is a closed L-subset, but the inverse is not true since every closed L-subset is not H-bounded set in general as the following example shows.

Example 3.12: By Example 3.9, let $\eta \in L^X$ be an L-subset, where $\eta =x_{.7}$ , then $\eta$ is closed L-subset because ${\tau }^{\prime }=\left\{0_X,x_{.7},x_{.3},1_X\right\}$ . But $x_{.7}\in L^X$ is not H-bounded set.

Theorem 3.13: Let $\left(L^X,\tau \right)$ be an L-ts. The following statements hold:

(i) $0_X,1_X\in HBC\left(L^X,\tau \right)$ .

(ii) If ${\mu }_1,{\mu }_2,\cdots ,{\mu }_n\in HBC\left(L^X,\tau \right)$ , then $\underset{i=1}{\overset{n}{\vee }}{\mu }_i\in HBC\left(L^X,\tau \right)$ .

(iii) If $\left\{{\mu }_i:i\in I\right\}\subseteq HBC\left(L^X,\tau \right)$ , then $\underset{i\in I}{\wedge }{\mu }_i\in HBC\left(L^X,\tau \right)$ .

(iv) Every H-bounded and closed set is HB-closed.

(v) $\mu \in L^X$ is HB-closed iff there exists $\lambda \in HB{R}_{x_{\alpha }}$ such that $\mu \le \lambda$ for

each $x_{\alpha }\in M\left(L^X\right)$ with $x_{\alpha }\notin \mu$

Proof. (i) Obvious.

(ii) Let ${\mu }_1,{\mu }_2,\cdots ,{\mu }_n\in HBC\left(L^X,\tau \right)$ and $x_{\alpha }\in M\left(L^X\right)$ such that

$x_{\alpha }\in HB.cl\left(\underset{i=1}{\overset{n}{\vee }}{\mu }_i\right)$ , then for each $\lambda \in HB{R}_{x_{\alpha }}$ we have $\underset{i=1}{\overset{n}{\vee }}{\mu }_i\cancel{\le }\lambda$ and so ${\mu }_i\cancel{\le }\lambda$

for some $i=1,2,\cdots ,n$ . Hence $x_{\alpha }\in HB.cl\left({\mu }_i\right)$ for some $i=1,2,\cdots ,n$ . Since ${\mu }_i$ is HB-closed set, then $HB.cl\left({\mu }_i\right)\le {\mu }_i$ for some $i=1,2,\cdots ,n$ and so $x_{\alpha }\in {\mu }_i$

for some $i=1,2,\cdots ,n$ and hence $x_{\alpha }\in \underset{i=1}{\overset{n}{\vee }}{\mu }_i$ . Thus $HB.cl\left(\underset{i=1}{\overset{n}{\vee }}{\mu }_i\right)\le \underset{i=1}{\overset{n}{\vee }}{\mu }_i$ ( ∗ )

Conversely, since ${\mu }_i\le HB.cl\left({\mu }_i\right)$ then $\underset{i=1}{\overset{n}{\vee }}{\mu }_i\le HB.cl\left(\underset{i=1}{\overset{n}{\vee }}{\mu }_i\right)$ ( $\ast$ $\ast$). Hence from ( $\ast$ ) and ( $\ast$ $\ast$) we have $HB.cl\left(\underset{i=1}{\overset{n}{\vee }}{\mu }_i\right)=\underset{i=1}{\overset{n}{\vee }}{\mu }_i$ . Thus $\underset{i=1}{\overset{n}{\vee }}{\mu }_i\in HBC\left(L^X,\tau \right)$ .

(iii) Let ${\mu }_1,{\mu }_2,\cdots ,{\mu }_n\in HBC\left(L^X,\tau \right)$ and $x_{\alpha }\in M\left(L^X\right)$ such that

$x_{\alpha }\in HB.cl\left(\underset{i\in I}{\wedge }{\mu }_i\right)$ , then for each $\lambda \in HB{R}_{x_{\alpha }}$ we have $\underset{i\in I}{\wedge }{\mu }_i\cancel{\le }\lambda$ and so ${\mu }_i\cancel{\le }\lambda$

for each $i\in I$ . Hence $x_{\alpha }\in HB.cl\left({\mu }_i\right)$ for each $i\in I$ . Since ${\mu }_i$ is HB-closed set, then $HB.cl\left({\mu }_i\right)\le {\mu }_i$ for each $i\in I$ and so $x_{\alpha }\in {\mu }_i$ for each $i\in I$ and

hence $x_{\alpha }\in \underset{i\in I}{\wedge }{\mu }_i$ . Thus $HB.cl\left(\underset{i\in I}{\wedge }{\mu }_i\right)\le \underset{i\in I}{\wedge }{\mu }_i$ ( $\ast$ ).

Conversely, since ${\mu }_i\le HB.cl\left({\mu }_i\right)$ then $\underset{i\in I}{\wedge }{\mu }_i\le HB.cl\left(\underset{i\in I}{\wedge }{\mu }_i\right)$ ( $\ast$ $\ast$). Hence from ( $\ast$ ) and ( $\ast$ $\ast$) we have $HB.cl\left(\underset{i\in I}{\wedge }{\mu }_i\right)=\underset{i\in I}{\wedge }{\mu }_i$ . Thus $\underset{i\in I}{\wedge }{\mu }_i\in HBC\left(L^X,\tau \right)$ .

(iv) Let $\mu \in L^X$ be an H-bounded and closed set and let $x_{\alpha }\in M\left(L^X\right)$ such

that $x_{\alpha }\notin \mu$ , since $\mu$ is H-bounded and closed set, then $\mu \in HB{R}_{x_{\alpha }}$ , since

$\mu \le \mu$ then $x_{\alpha }\notin HB.cl\left(\mu \right)$ and so $HB.cl\left(\mu \right)\le \mu$ . Therefore $\mu$ is HB-closed set.

(v) Suppose that $\mu$ is HB-closed set, $x_{\alpha }\in M\left(L^X\right)$ and $x_{\alpha }\notin \mu$ . By Definition 3.9, there exists $\lambda \in HB{R}_{x_{\alpha }}$ with $\mu \le \lambda$ . Conversely, provided that the condition is satisfied. If $\mu$ is not HB-closed set, then there exists $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in HB.cl\left(\mu \right)$ and $x_{\alpha }\notin \mu$ . Hence $\mu \cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ . It

conflicts with the hypothesis, and so $\mu$ is HB-closed set.

Theorem 3.14: Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then $\mu \in HBC\left(L^X,\tau \right)$ iff $\mu \in HB{R}_{x_{\alpha }}$ for each $x_{\alpha }\notin \mu$ .

Proof. It follows directly from Theorem 3.13 (v).

Theorem 3.15: Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then the mapping $HB.cl:L^X\to L^X$ is called closure operator of HB-boundedness iff it satisfies:

(i) $HB.cl\left(0_X\right)=0_X$ .

(ii) $\mu \le HB.cl\left(\mu \right)$ .

(iii) $HB.cl\left(\mu \vee \eta \right)=HB.cl\left(\mu \right)\vee HB.cl\left(\eta \right)$ .

(iv) $HB.cl\left(HB.cl\left(\mu \right)\right)=HB.cl\left(\mu \right)$.

A closure operator of HB-boundedness $HB.cl$ generates L-topology ${\tau }_{HB.cl}$ on $L^X$ as: ${\tau }_{HB.cl}=\left\{\mu \in L^X:HB.cl\left({\mu }^{\prime }\right)={\mu }^{\prime }\right\}$ .

Proof. It follows directly from Theorems 3.11 and 3.13.

Theorem 3.16: Let $\left(L^X,\tau \right)$ be an L-ts. Then:

(i) ${\tau }_{HB}\le \tau$ .

(ii) If $\left(L^X,\tau \right)$ is H-bounded space, then $\tau ={\tau }_{HB}$ .

Proof. (i) Let $\mu \in {\tau }_{HB}$ , then $HB.cl\left({\mu }^{\prime }\right)\le {\mu }^{\prime }$ . Since $cl\left({\mu }^{\prime }\right)\le HB.cl\left({\mu }^{\prime }\right)$ , hence

$cl\left({\mu }^{\prime }\right)\le {\mu }^{\prime }$ and so $\mu \in \tau$ .

(ii) We note that ${\tau }_{HB}\le \tau$ from (i). Now, let $\mu \in \tau$ then ${\mu }^{\prime }\in {\tau }^{\prime }$ . Since $1_X$ is H-bounded and ${\mu }^{\prime }\le 1_X$ , then ${\mu }^{\prime }$ is H-bounded (By Theorem 3.3 (ii)) and by Theorem 3.13 (iv) we have ${\mu }^{\prime }$ is HB-closed set and so ${\mu }^{\prime }\in {\tau }_{HB}$ . Thus $\tau ={\tau }_{HB}$ .

Definition 3.17. Let $\left(L^X,\tau \right)$ be an L-ts, $\mu \in L^X$ and

$HB.\mathrm{int}\left(\mu \right)=\vee \left\{\rho \in L^X:\rho \in HBO\left(L^X,\tau \right),\rho \le \mu \right\}$ . We say that $HB.\mathrm{int}\left(\mu \right)$ is the HB-interior of $\mu$ .

The following Theorem shows the relationships between HB-closure operator and HB-interior operator.

Theorem 3.18: Let $\left(L^X,\tau \right)$ be an L-ts and $\mu \in L^X$ . Then the following are true:

(i) $\mu$ is HB-open iff $\mu =HB.\mathrm{int}\left(\mu \right)$ .

(ii) ${\left(HB.cl\left(\mu \right)\right)}^{\prime }=HB.\mathrm{int}\left({\mu }^{\prime }\right)$ and ${\left(HB.\mathrm{int}\left(\mu \right)\right)}^{\prime }=HB.cl\left({\mu }^{\prime }\right)$ .

(iii) $HB.cl\left(\mu \right)={\left(HB.\mathrm{int}\left({\mu }^{\prime }\right)\right)}^{\prime }$ and $HB.\mathrm{int}\left(\mu \right)={\left(HB.cl\left({\mu }^{\prime }\right)\right)}^{\prime }$ .

(iv) $HB.\mathrm{int}\left(\mu \right)\le \mathrm{int}\left(\mu \right)\le \mu$.

(v) If $\eta \in L^X$ and $\mu \le \eta$ then $HB.\mathrm{int}\left(\mu \right)\le HB.\mathrm{int}\left(\eta \right)$ .

(vi) $HB.\mathrm{int}\left(HB.\mathrm{int}\left(\mu \right)\right)=HB.\mathrm{int}\left(\mu \right)$.

Proof. (i) Let $\mu \in L^X$ be an HB-open set, then

$HB.\mathrm{int}\left(\mu \right)=\vee \left\{\rho \in L^X:\rho \in HBO\left(L^X,\tau \right),\rho \le \mu \right\}=\mu$ and so $\mu =HB.\mathrm{int}\left(\mu \right)$ .

Conversely, let $\mu =HB.\mathrm{int}\left(\mu \right)$ , since

$HB.\mathrm{int}\left(\mu \right)=\vee \left\{\rho \in L^X:\rho \in HBO\left(L^X,\tau \right),\rho \le \mu \right\}$ . Therefore $\mu$ is HB-open set.

(ii) It follows directly from Definition 3.17 and Theorem 3.11 (iv).

(iii) It follows directly from (ii)

(iv) It follows directly from (ii) and Theorems 3.11 (i)

(v) It follows directly from (ii) and Theorem 3.11 (ii)

(vi) It follows directly from (ii) and Theorem 3.11 (iii)

Theorem 3.19: Let $\left(L^X,\tau \right)$ be an L-ts. The following statements hold::

(i) $0_X,1_X\in HBO\left(L^X,\tau \right)$ .

(ii) If ${\mu }_1,{\mu }_2,\cdots ,{\mu }_n\in HBO\left(L^X,\tau \right)$ , then $\underset{i=1}{\overset{n}{\wedge }}{\mu }_i\in HBO\left(L^X,\tau \right)$ .

(iii) If $\left\{{\mu }_i:i\in I\right\}\subseteq HBO\left(L^X,\tau \right)$ , then $\underset{i\in I}{\vee }{\mu }_i\in HBO\left(L^X,\tau \right)$ .

Definition 3.20: Let $\left(L^X,\tau \right)$ be an L-ts and S be a molecular net in $L^X$ . Then $x_{\alpha }\in M\left(L^X\right)$ is called

(i) limit point of S [13] , (or S converges to $x_{\alpha }$ ) in symbol $S\to x_{\alpha }$ if for

every $\mu \in R_{x_{\alpha }}$ there is $n\in D$ such for each $m\in D$ and $m\ge n$ we have

$S\left(m\right)\notin \mu$ . The union of all limit points of S is denoted by $\lim \left(S\right)$ .

(ii) H-bounded limit point of S, (or S HB-converges to $x_{\alpha }$ ) in symbol

$S\stackrel{HB}{\to }{x}_{\alpha }$ if for every $\mu \in HB{R}_{x_{\alpha }}$ there is an $n\in D$ such that $m\in D$ and

$m\ge n$ , we have $S\left(m\right)\notin \mu$ . The union of all HB-limit points of S is denoted by $HB.\lim \left(S\right)$ .

Theorem 3.21: Suppose that S is a molecular net in $\left(L^X,\tau \right)$ , $\mu \in L^X$ and $x_{\alpha }\in M\left(L^X\right)$ . Then the following statements hold:

(i) If $S\to x_{\alpha }$ , then $S\stackrel{HB}{\to }{x}_{\alpha }$ .

(ii) $x_{\alpha }\in HB.\lim \left(S\right)$ iff $S\stackrel{HB}{\to }{x}_{\alpha }$ .

(iii) $\lim \left(S\right)\le HB.\lim \left(S\right)$ .

(iv) $x_{\alpha }\in HB\mathrm{..}cl\left(\mu \right)$ (resp. $x_{\alpha }\in .cl\left(\mu \right)$ ), iff there exists a molecular net S in $\mu$ such that S is HB-converges (resp. converges) to $x_{\alpha }$ .

(v) $HB.\lim \left(S\right)$ is HB-closed set in $L^X$ .

Proof. (i) Let $S\to x_{\alpha }$ and let $\lambda \in HB{R}_{x_{\alpha }}$ . Since $HB{R}_{x_{\alpha }}\subseteq R_{x_{\alpha }}$ , then $\lambda \in R_{x_{\alpha }}$ Since $S\to x_{\alpha }$ , then for every $\mu \in R_{x_{\alpha }}$ there is $n\in D$ such for each $m\in D$ and $m\ge n$ , we have $S\left(m\right)\notin \lambda$ . Thus $S\stackrel{HB}{\to }{x}_{\alpha }$ .

(ii) Let $x_{\alpha }\in HB.\lim \left(S\right)$ and let $\lambda \in HB{R}_{x_{\alpha }}$ . Since $x_{\alpha }\notin \lambda$ , then

$HB.\lim \left(S\right)\notin \lambda$ . Therefore there exists $y_{\gamma }\in M\left(L^X\right)$ such that

$y_{\gamma }\in HB.\lim \left(S\right)$ and $y_{\gamma }\notin \lambda$ . Then $\lambda \in HB{R}_{y_{\gamma }}$ and so there is $n\in D$ much

for each $m\in D$ and $m\ge n$ we have $S\left(m\right)\notin \lambda$ , but since $\lambda \in HB{R}_{x_{\alpha }}$ so $S\stackrel{HB}{\to }{x}_{\alpha }$ . Conversely, let $S\stackrel{HB}{\to }{x}_{\alpha }$ , then by Definition 3.20 (ii) we have

$x_{\alpha }\in HB.\lim (\; S\; )$

(iii) Let $x_{\alpha }\in \lim \left(S\right)$ and let $\eta \in HB{R}_{x_{\alpha }}$ . Since $HB{R}_{x_{\alpha }}\subseteq R_{x_{\alpha }}$ , then $\eta \in R_{x_{\alpha }}$ . And since $x_{\alpha }\in \lim \left(S\right)$ , then for each $\lambda \in R_{x_{\alpha }}$ there is $n\in D$ such for each

$m\in D$ and $m\ge n$ , we have $S\left(m\right)\notin \lambda$ and so $S\left(m\right)\notin \eta$ . Hence

$x_{\alpha }\in HB.\lim \left(S\right)$ . So $\lim \left(S\right)\le HB.\lim \left(S\right)$ .

(iv) Let $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in HB.cl\left(\mu \right)$ , then $\mu \cancel{\le }\lambda$ for each

$\lambda \in HB{R}_{x_{\alpha }}$ . Since $\mu \cancel{\le }\lambda$ , then there exists $\alpha \left(\mu ,\lambda \right)\in M\left(L\right)$ such that $x_{\alpha \left(\mu ,\lambda \right)}\in \mu$ with $x_{\alpha \left(\mu ,\lambda \right)}\notin \lambda$ . Since the pair $\left(HB{R}_{x_{\alpha }},\ge \right)$ is a directed set and so we can define a molecular net $S:HB{R}_{x_{\alpha }}\to M\left(L^X\right)$ as follows $S\left(\lambda \right)=x_{\alpha \left(\mu ,\lambda \right)}$ for each $\lambda \in HB{R}_{x_{\alpha }}$ Hence S is a molecular net in $\mu$ . Now let $\eta \in HB{R}_{x_{\alpha }}$

such that $\lambda \le \eta$ , so we have there exists $S\left(\eta \right)=x_{\alpha \left(\mu ,\eta \right)}\notin \eta$ and so

$S\left(\eta \right)=x_{\alpha \left(\mu ,\eta \right)}\notin \lambda$ . Hence S is HB-converges to $x_{\alpha }$ .

Conversely, let S be a molecular net in $\mu$ such that S is HB-converges to $x_{\alpha }$

then for each $\lambda \in HB{R}_{x_{\alpha }}$ there is $n\in D$ such for each $m\in D$ and $m\ge n$ ,

we have $S\left(m\right)\notin \lambda$ . Since $S\left(n\right)\in \mu$ for each $n\in D$ , $m\in D$ . So $S\left(m\right)\in \mu$

and $\mu \ge S\left(m\right)>\lambda$ hence $\mu \cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ . This means that

$x_{\alpha }\in HB.cl\left(\mu \right)$ .

(v) Let $x_{\alpha }\in HB.cl\left(HB.\lim \left(S\right)\right)$ , then $HB.\lim \left(S\right)\cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ and then there exists $y_{\gamma }\in M\left(L^X\right)$ such that $y_{\gamma }\in HB.\lim \left(S\right)$ and $y_{\gamma }\notin \lambda$ . Then for each $\mu \in HB{R}_{y_{\gamma }}$ , there is $n\in D$ much for each $m\in D$ and $m\ge n$

we have $S\left(m\right)\notin \mu$ and so $S\left(m\right)\notin \lambda$ . Hence $x_{\alpha }\in HB.\lim \left(S\right)$ . Thus

$HB.cl\left(HB.\lim \left(S\right)\right)\le HB.\lim \left(S\right)$ and so $HB.\lim \left(S\right)$ is HB-closed set.

Definition 3.22: Let $\left(L^X,\tau \right)$ be an L-ts and I be an ideal in $L^X$ . Then $x_{\alpha }\in M\left(L^X\right)$ is called:

(i) limit point of I [18] , (or I converges to $x_{\alpha }$ ) in symbol $I\to x_{\alpha }$ if $R_{x_{\alpha }}\subseteq I$ . The union of all limit points of I is denoted by $\lim \left(I\right)$ .

(ii) H-bounded limit point of I, (or I HB-converges to $x_{\alpha }$ ) in symbol $I\stackrel{HB}{\to }{x}_{\alpha }$ if $HB{R}_{x_{\alpha }}\subseteq I$ . The union of all HB-limit points of I is denoted by $HB.\lim \left(I\right)$ .

Theorem 3.23: Suppose that I is an ideal in $\left(L^X,\tau \right)$ , $\mu \in L^X$ and $x_{\alpha }\in M\left(L^X\right)$ . Then the following statements hold:

(i) If $I\to x_{\alpha }$ , then $I\stackrel{HB}{\to }{x}_{\alpha }$ .

(ii) $x_{\alpha }\in HB.\lim \left(I\right)$ iff $I\stackrel{HB}{\to }{x}_{\alpha }$ .

(iii) $\lim \left(I\right)\le HB.\lim \left(I\right)$ .

(iv) $x_{\alpha }\in HB\mathrm{..}cl\left(\mu \right)$ iff there exists an ideal I in $L^X$ such that $I\stackrel{HB}{\to }{x}_{\alpha }$ and $\mu \notin I$

(v) $HB.\lim \left(I\right)$ is HB-closed set in $L^X$ .

Proof. (i) Let $I\to x_{\alpha }$ then $R_{x_{\alpha }}\subseteq I$ . Since $HB{R}_{x_{\alpha }}\subseteq R_{x_{\alpha }}$ , then $HB{R}_{x_{\alpha }}\subseteq I$ . Thus $I\stackrel{HB}{\to }{x}_{\alpha }$ .

(ii) Let $x_{\alpha }\in HB.\lim \left(I\right)$ and let $\lambda \in HB{R}_{x_{\alpha }}$ . Since $x_{\alpha }\notin \lambda$ and

$x_{\alpha }\in HB.\lim \left(I\right)$ , then $HB.\lim \left(I\right)\notin \lambda$ . Therefore there exists $y_{\gamma }\in M(\; L\; X\; )$

such that $y_{\gamma }\in HB.\lim \left(I\right)$ and $y_{\gamma }\notin \lambda$ . Then $\lambda \in HB{R}_{y_{\gamma }}$ and so

$HB{R}_{x_{\alpha }}\subseteq HB{R}_{y_{\gamma }}\subseteq I$ hence $HB{R}_{x_{\alpha }}\subseteq I$ . Thus $I\stackrel{HB}{\to }{x}_{\alpha }$ . Conversely, let $I\stackrel{HB}{\to }{x}_{\alpha }$ , then by Definition 3.22 (ii) we have $x_{\alpha }\in HB.\lim \left(I\right)$ .

(iii) Let $x_{\alpha }\in \lim \left(I\right)$ and let $\eta \in HB{R}_{x_{\alpha }}$ . since $x_{\alpha }\in \lim \left(I\right)$ , so for each $\lambda \in R_{x_{\alpha }}$ , $\lambda \in I$ and since $\eta \in HB{R}_{x_{\alpha }}$ so $\eta \in R_{x_{\alpha }}$ . Hence $x_{\alpha }\in HB.\lim \left(I\right)$ . So $\lim \left(I\right)\le HB.\lim \left(I\right)$ .

(iv) Let $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in HB.cl\left(\mu \right)$ . The family

$I=\left\{\rho \in L^X:\exists \lambda \in HB{R}_{x_{\alpha }}∍\rho \le \lambda \right\}$ is an ideal in $L^X$ . Now we show that $\mu \notin I$ . Since $x_{\alpha }\in HB.cl\left(\mu \right)$ , then for each $\lambda \in HB{R}_{x_{\alpha }}$ , $\mu \cancel{\le }\lambda$ . So By definition of I we have $\mu \notin I$ . Finally, we show that $I\stackrel{HB}{\to }{x}_{\alpha }$ . Let $\lambda \in HB{R}_{x_{\alpha }}$ , since $\lambda \le \lambda$ , then $\lambda \in I$ . So $HB{R}_{x_{\alpha }}\subseteq I$ . Thus $I\stackrel{HB}{\to }{x}_{\alpha }$ .

Conversely, let I be an ideal in $L^X$ such that $I\stackrel{HB}{\to }{x}_{\alpha }$ and $\mu \notin I$ . Then for each $\lambda \in HB{R}_{x_{\alpha }}$ , $\lambda \in I$ . Since $\lambda \in I$ , $\mu \notin I$ , then $\mu \cancel{\le }\lambda$ and so

$x_{\alpha }\in HB\mathrm{..}cl\left(\mu \right)$ .

(v) Let $x_{\alpha }\in HB\mathrm{..}cl\left(HB.\lim \left(I\right)\right)$ , then $HB.\lim \left(I\right)\cancel{\le }\lambda$ for each $\lambda \in HB{R}_{x_{\alpha }}$ and then there exists $y_{\gamma }\in M\left(L^X\right)$ such that $y_{\gamma }\in HB.\lim \left(I\right)$ and $y_{\gamma }\notin \lambda$ . Since $\lambda \in HB{R}_{y_{\gamma }}$ and $I\stackrel{HB}{\to }{y}_{\gamma }$ then $\eta \in I$ for each $\eta \in HB{R}_{x_{\alpha }}$ . Since $y_{\gamma }\notin \lambda$

then $\lambda \in I$ . But $\lambda \in HB{R}_{x_{\alpha }}$ and so $x_{\alpha }\in HB.\lim \left(I\right)$ . Thus

$HB\mathrm{..}cl\left(HB.\lim \left(I\right)\right)\le HB.\lim \left(I\right)$ and so $HB.\lim \left(I\right)$ is HB-closed set.

4. HB-Continuous Mappings in L-Topological Space

In this section we first define HB-continuous mappings in L-topological space and then investigate some of its characterizations,

Definition 4.1: An L-mapping $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is called :

(i) HB-continuous at $x_{\alpha }\in M\left(L^X\right)$ if $f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ for each $\eta \in HB{R}_{f_L(\; x\; \alpha \; )}$

(ii) HB-continuous if $f_L^{-1}\left(\eta \right)\in \tau$ for each $\eta \in L^X$ is closed and H-bounded.

Theorem 4.2: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-continuous mapping. Then the following properties are equivalent :

(i) $f_L$ is HB-continuous.

(ii) $f_L$ is HB-continuous at $x_{\alpha }$ for each $x_{\alpha }\in M\left(L^X\right)$ .

(iii) If $\eta \in \Delta$ and ${\eta }^{\prime }$ is H-bounded, then $f_L^{-1}\left(\eta \right)\in \tau$ .

(iv) If $\eta \in L^Y$ is H-bounded, then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ .

Proof. (i) $\Rightarrow$ (ii): Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an HB-continuous and

$x_{\alpha }\in M\left(L^X\right)$ , $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Since $f_L\left(x_{\alpha }\right)\notin \eta$ , then

$x_{\alpha }\notin f_L^{-1}(\; \eta \; )$

And so $f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ . Thus $f_L$ is HB-continuous at $x_{\alpha }$ for each

$x_{\alpha }\in M\left(L^X\right)$ .

(ii) $\Rightarrow$ (i): Let $f_L$ be an HB-continuous at $x_{\alpha }$ for each $x_{\alpha }\in M\left(L^X\right)$ . If $f_L$ is not HB-continuous, then there is $\eta \in L^Y$ is H-bounded and closed such that $f_L^{-1}\left(\eta \right)\notin {\tau }^{\prime }$ , i.e., $cl\left(f_L^{-1}\left(\eta \right)\right)\cancel{\le }{f}_L^{-1}\left(\eta \right)$ . Then there exists $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in cl\left(f_L^{-1}\left(\eta \right)\right)$ and $x_{\alpha }\notin f_L^{-1}\left(\eta \right)$ implies that $f_L\left(x_{\alpha }\right)\notin \eta$ , since $\eta$ is

closed and H-bounded, then $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ . But $f_L^{-1}\left(\eta \right)\notin R_{x_{\alpha }}$ , this contradiction. Thus $f_L$ is HB-continuous mapping.

(i) $\Rightarrow$ (iii): Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an HB-continuous and $\eta \in \Delta$ such that ${\eta }^{\prime }$ is H-bounded and so ${\eta }^{\prime }$ is H-bounded and closed. By (i), we have $f_L^{-1}\left({\eta }^{\prime }\right)\in {\tau }^{\prime }$ . Since $f_L^{-1}\left({\eta }^{\prime }\right)={\left(f_L^{-1}\left(\eta \right)\right)}^{\prime }$ , then $f_L^{-1}\left(\eta \right)\in \tau$ .

(iii) $\Rightarrow$ (i): Let $\eta \in L^Y$ be an H-bounded and closed, then ${\eta }^{\prime }\in \Delta$ . By (iii), we have $f_L^{-1}\left({\eta }^{\prime }\right)\in \tau$ , thus $f_L^{-1}\left(\eta \right)={\left(f_L^{-1}\left({\eta }^{\prime }\right)\right)}^{\prime }$ , then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Hence $f_L$ is HB-continuous mapping.

(iv) $\Rightarrow$ (iii): Let $\eta \in \Delta$ and ${\eta }^{\prime }$ be an H-bounded. By (iv), we have $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L^{-1}\left(\eta \right)={\left(f_L^{-1}\left({\eta }^{\prime }\right)\right)}^{\prime }\in \tau$ .

(iv) $\Rightarrow$ (ii): Let $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ and $x_{\alpha }\in M\left(L^X\right)$ . Then $\eta$ is closed and H-bounded set, $f_L\left(x_{\alpha }\right)\notin \eta$ and so $x_{\alpha }\notin f_L^{-1}\left(\eta \right)$ . By (iv), we have $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ and $x_{\alpha }\notin f_L^{-1}\left(\eta \right)$ hence $f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ . Thus $f_L$ is HB-continuous mapping at $x_{\alpha }$ for each $x_{\alpha }\in M\left(L^X\right)$ .

(iv) $\Rightarrow$ (i): Let $\eta \in L^Y$ be a closed and H-bounded set. By (iv), we have $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is HB-continuous mapping.

Theorem 4.3: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-surjective mapping. Then the following conditions are equivalent:

(i) $f_L$ is HB-continuous mapping.

(ii) For each $\mu \in L^X$ , $f_L\left(cl\left(\mu \right)\right)\le HB.cl\left(f_L\left(\mu \right)\right)$ ,

(iii) For each $\eta \in L^Y$ , $cl\left(f_L^{-1}\left(\eta \right)\right)\le f_L^{-1}\left(HB.cl\left(\eta \right)\right)$ ,

(iv) For each $\eta \in L^Y$ , $f_L^{-1}\left(HB.\mathrm{int}\left(\eta \right)\right)\le \mathrm{int}\left(f_L^{-1}\left(\eta \right)\right)$ ,

(v) For each HB-open L-subset $\rho$ in $L^Y$ , then $f_L^{-1}\left(\rho \right)$ is open L-subset in $L^X$ ,

(vi) For each HB-closed L-subset $\lambda$ in $L^Y$ , then $f_L^{-1}\left(\lambda \right)$ is closed L-subset in $L^X$ .

Proof. (i) $\Rightarrow$ (ii): Let $\mu \in L^X$ and $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in cl\left(\mu \right)$ . Then

$f_L\left(x_{\alpha }\right)\in f_L\left(cl\left(\mu \right)\right)$ . Let $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ . So by (i) and by Theorem 4.3, we have $f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ . Since $x_{\alpha }\in cl\left(\mu \right)$ , then $\mu \cancel{\le }{f}_L^{-1}\left(\eta \right)$ . Since $f_L$ is L-surjective then $f_L\left(\mu \right)\cancel{\le }\eta$ and $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ so $f_L\left(x_{\alpha }\right)\in HB.cl\left(f_L\left(\mu \right)\right)$ . Hence $f_L\left(cl\left(\mu \right)\right)\le HB.cl\left(f_L\left(\mu \right)\right)$ .

(ii) $\Rightarrow$ (iii): Let $\eta \in L^Y$ . Then $f_L^{-1}\left(\eta \right)\in L^X$ . By (ii) we have

$f_L\left(cl\left(f_L^{-1}\left(\eta \right)\right)\right)\le HB.cl\left(f_L\left(f_L^{-1}\left(\eta \right)\right)\right)\le HB.cl\left(\eta \right)$ . So

$f_L\left(cl\left(f_L^{-1}\left(\eta \right)\right)\right)\le HB.cl\left(\eta \right)$ . Thus $f_L^{-1}{f}_L\left(cl\left(f_L^{-1}\left(\eta \right)\right)\right)\le f_L^{-1}\left(HB.cl\left(\eta \right)\right)$ . Since $cl\left(f_L^{-1}\left(\eta \right)\right)\le f_L^{-1}{f}_L\left(cl\left(f_L^{-1}\left(\eta \right)\right)\right)$ , then $cl\left(f_L^{-1}\left(\eta \right)\right)\le f_L^{-1}\left(HB.cl\left(\eta \right)\right)$ .

(iii) $\Rightarrow$ (iv): Let $\eta \in L^Y$ . By (iii), we have $cl\left(f_L^{-1}\left({\eta }^{\prime }\right)\right)\le f_L^{-1}\left(HB.cl\left({\eta }^{\prime }\right)\right)$ Since $cl\left(f_L^{-1}\left({\eta }^{\prime }\right)\right)={\left(\mathrm{int}\left(f_L^{-1}\left(\eta \right)\right)\right)}^{\prime }$ and $f_L^{-1}\left(HB.cl\left({\eta }^{\prime }\right)\right)={\left(f_L^{-1}\left(HB.\mathrm{int}\left(\eta \right)\right)\right)}^{\prime }$ . So ${\left(\mathrm{int}\left(f_L^{-1}\left(\eta \right)\right)\right)}^{\prime }\le {\left(f_L^{-1}\left(HB.\mathrm{int}\left(\eta \right)\right)\right)}^{\prime }$ . Thus $f_L^{-1}\left(HB.\mathrm{int}\left(\eta \right)\right)\le \mathrm{int}\left(f_L^{-1}\left(\eta \right)\right)$ .

(iv) $\Rightarrow$ (v): Let $\rho$ be an HB-open L-subset in $L^Y$ . Then

$f_L^{-1}\left(\rho \right)=f_L^{-1}\left(HB.\mathrm{int}\left(\rho \right)\right)$ and by (iv), we have

$f_L^{-1}\left(HB.\mathrm{int}\left(\rho \right)\right)\le \mathrm{int}\left(f_L^{-1}\left(\rho \right)\right)$ , so $f_L^{-1}\left(\rho \right)\le \mathrm{int}\left(f_L^{-1}\left(\rho \right)\right)$ . Thus $f_L^{-1}\left(\rho \right)\in \tau$ .

(v) $\Rightarrow$ (vi): Let $\lambda$ be an HB-closed L-subset in $L^Y$ . By (v), we have $f_L^{-1}\left({\lambda }^{\prime }\right)\in \tau$ . Then ${\left(f_L^{-1}\left(\lambda \right)\right)}^{\prime }=f_L^{-1}\left({\lambda }^{\prime }\right)\in \tau$ and so $f_L^{-1}\left(\lambda \right)\in {\tau }^{\prime }$ .

(vi) $\Rightarrow$ (i): Let $\eta \in L^Y$ be an closed and H-bounded set, then $\eta$ is HB-closed L-subset in $L^Y$ . By (vi), we have $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is HB-continuous mapping.

Theorem 4.4: If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping, then

$f_L:\left(L^X,\tau \right)\to \left(L^{f\left(X\right)},{\Delta }_{f\left(X\right)}\right)$ is HB-continuous mapping.

Proof. Let $\eta \in {\Delta }_{f\left(X\right)}$ such that $1_{f\left(X\right)}\backslash \eta$ is H-bounded set, then $1_{f\left(X\right)}\backslash \eta$ is

H-bounded and closed in $\left(L^{f\left(X\right)},{\Delta }_{f\left(X\right)}\right)$ . Therefore $\rho =1_Y\backslash \left(1_{f\left(X\right)}\backslash \eta \right)\in \Delta$ and

${\rho }^{\prime }$ is H-bounded in $\left(L^Y,\Delta \right)$ . Since $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping, the by Theorem 4.2 (iii), we have $f_L^{-1}\left(\rho \right)\in \tau$ , thus

$f_L^{-1}\left(\rho \right)=f_L^{-1}\left(1_Y\backslash \left(1_{f\left(X\right)}\backslash \eta \right)\right)=1_X\backslash \left(f_L^{-1}\left(1_{f\left(X\right)}\backslash \eta \right)\right)=1_X\backslash \left(1_X\backslash f_L^{-1}\left(\eta \right)\right)=f_L^{-1}\left(\eta \right)$ .

Hence $f_L^{-1}\left(\eta \right)\in \tau$ consequently, $f_L:\left(L^X,\tau \right)\to \left(L^{f\left(X\right)},{\Delta }_{f\left(X\right)}\right)$ is

HB-continuous mapping.

Theorem 4.5: If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping and

$A\subseteq X$ then ${f_L|}_A:\left(L^A,{\tau }_A\right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping.

Proof. Let $\eta \in L^Y$ be an H-bounded and closed set. Since

$f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping, then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ and since

${\left({f_L|}_A\right)}^{-1}\left(\eta \right)=f_L^{-1}\left(\eta \right)\wedge 1_A\in {{\tau }^{\prime }}_A$ . Hence ${f_L|}_A$ is HB-continuous mapping.

Theorem 4.6: Every $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ L-continuous mapping is HB-continuous mapping.

Proof. Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-continuous and let $\eta \in L^Y$ be an closed and H-bounded set, then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is HB-continuous mapping.

The following example shows that the converse is not true in general.

Example 4.7: Let $\left\{I_j:j\in J\right\}$ be the usual interval base of the relative L-topology on $L=I=\left[0,1\right]$ induced by the set of real numbers. Define a L-topology $\tau$ on $\left[0,1\right]$ generated by the base consisting of, $0_X$ , $1_X$ and $\left\{I_{j_k}:j\in J\text{and}k\in \left(0,1\right)\right\}$ where

$I_{j_K}\left(x\right)=\{\begin{array}{l}k:x\in I\\ 0:x\notin I\end{array}$

Let $\Delta$ be the L-topology on I such that the complements of any number of $\Delta$ is countable L-subset in I (i.e., the support of the L-subset is countable). Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be a function defined by $f\left(x\right)=x$ , for all $x\in I$ . Then it can be see that $f_L$ is HB-continuous but not L-continuous mapping.

Theorem 4.8: A mapping $f_L:\left(L^X,\tau \right)\to \left(L^Y,{\Delta }_{HB}\right)$ is L-continuous mapping iff it is HB-continuous mapping.

Proof. Since ${{\Delta }^{\prime }}_{HB}\le {\Delta }^{\prime }$ , then necessity is evident. Now, we suppose that $f_L$ is HB-continuous and $\eta \in {{\Delta }^{\prime }}_{HB}$ . Then by Theorem 4.3 (iii) we have $f_L^{-1}\left(\eta \right)=f_L^{-1}\left(HB.cl\left(\eta \right)\right)\ge cl\left(f_L^{-1}\left(\eta \right)\right)$ and so $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is L-continuous mapping.

Theorem 4.9: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-mapping and $\left(L^Y,\Delta \right)$ is H-bounded space. Then $f_L$ is L-continuous mapping iff $f_L$ is HB-continuous mapping.

Proof. By Theorem 4.6 we need only to investigate the sufficiency. Let $\eta \in {\Delta }^{\prime }$ . Since $\left(L^Y,\Delta \right)$ is H-bounded space then by Theorem 3.2(ii), we have $\eta$ is H-bounded set and so $\eta$ is HB-closed L-subset. By HB-continuity of $f_L$ , we have $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Hence $f_L$ is L-continuous mapping.

Theorem 4.10: If $f_L$ is HB-continuous, then $f_L$ is H-continuous mapping.

Proof. Follows from the fact that every H-compact set is H-bounded set.

Theorem 4.11: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an L-mapping and $\left(L^Y,\Delta \right)$ be $L{T}_3$ -space. Then $f_L$ is H-continuous iff $f_L$ is HB-continuous mapping.

Proof. Let $f_L$ be an HB-continuous mapping and let $\eta \in L^Y$ be a closed and H-compact, then by Theorem 3.3 (i), we have $\eta$ is H-bounded and closed. Since $f_L$ is HB-continuous then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is H-continuous.

Conversely, let $f_L$ be an H-continuous and let $\eta \in L^Y$ be a closed and H-bounded. Then $\eta$ is H-compact and closed. Since $f_L$ is H-continuous, then $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is HB-continuous mapping.

Remark 4.12: For an L-mapping $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ , we obtain the following implications:

L-continuity HB-continuity H-continuity.

None of these implications are reversible. However, if it $\left(L^Y,\Delta \right)$ is H-bounded (resp. $L{T}_3$ -) space, then Theorem 4.10 (resp. Theorem 4.12) implies that the concepts of L-continuity (resp. HB-continuity) and H-continuity are equivalent.

Theorem 4.13: If $f_L:\left(L^X,{\tau }_1\right)\to \left(L^Y,{\tau }_2\right)$ is L-continuous and

$g_L:\left(L^Y,{\tau }_2\right)\to \left(L^Z,{\tau }_3\right)$ is HB-continuous, then $g_L\circ f_L:\left(L^X,{\tau }_1\right)\to \left(L^Z,{\tau }_3\right)$ is HB-continuous.

Proof. Let $\eta \in L^Y$ be a closed and almost N-compact. Since $g_L$ is HB-continuous, then $g_L^{-1}\left(\eta \right)\in {{\tau }^{\prime }}_2$ and since $f_L$ is L-continuous, then

$f_L^{-1}\left(g_L^{-1}\left(\eta \right)\right)\in {{\tau }^{\prime }}_1$ Hence $g_L\circ f_L$ HB-continuous mapping.

Theorem 4.14: If $\left(L^X,\tau \right)$ and $\left(L^Y,\Delta \right)$ are L-ts's and $1_X=1_A\vee 1_B$ such

that $1_A,1_B\in {\tau }^{\prime }$ and $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is L-mapping and ${f_L|}_A,{f_L|}_B$ are

HB-continuous mappings, then $f_L$ is HB-continuous mapping.

Proof. Let $\eta \in L^Y$ be an N-almost bounded and closed then

$\begin{array}{c}{\left({f_L|}_A\right)}^{-1}\left(\eta \right)\vee {\left({f_L|}_B\right)}^{-1}\left(\eta \right)=\left(f_L^{-1}\left(\eta \right)\wedge 1_A\right)\vee \left(f_L^{-1}\left(\eta \right)\wedge 1_B\right)\\ =\left(f_L{}^{-1}\left(\eta \right)\wedge \left(1_A\vee 1_B\right)\right)=f_L^{-1}\left(\eta \right)\wedge 1_X=f_L^{-1}(\; \eta \; )\end{array}$

Hence $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Thus $f_L$ is HB-continuous mapping.

Theorem 4.15: If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping, injective, $\left(L^Y,\Delta \right)$ is $L{T}_1$ -space and H-bounded, then $\left(L^X,\tau \right)$ is $L{T}_1$ -space.

Proof. Let $x_{\alpha },y_{\gamma }\in M\left(L^X\right)$ such that $x\ne y$ . Since $f_L$ is injective L-mapping, then $f_L\left(x_{\alpha }\right),f_L\left(y_{\gamma }\right)\in M\left(L^Y\right)$ and $f\left(x\right)\ne f\left(y\right)$ . Since $\left(L^Y,\Delta \right)$ is $L{T}_1$ -space, then $f_L\left(x_{\alpha }\right),f_L\left(y_{\gamma }\right)$ are closed L-subsets in $\left(L^Y,\Delta \right)$ . Since $\left(L^Y,\Delta \right)$ is H-bounded, then $f_L\left(x_{\alpha }\right),f_L\left(y_{\gamma }\right)$ are H-bounded L-subsets. Since $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is HB-continuous mapping, then $f_L^{-1}\left(f_L\left(x_{\alpha }\right),\right)=x_{\alpha }$ and $f_L^{-1}\left(f_L\left(y_{\gamma }\right),\right)=y_{\gamma }$ are closed L-subsets in $\left(L^X,\tau \right)$ . Hence $\left(L^X,\tau \right)$ is $L{T}_1$ -space.

5. Characterizations of HB-Continuous Mappings in L-Topological Space

Theorem 5.1: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be an HB-continuous mapping and

be a fully stratified $L{T}_{2\frac{1}{2}}$ -space and $L{R}_2$ -space. If $f_L\left(1_X\right)$ is contained in

some H-compact set of $L^Y$ , then $f_L$ is L-continuous mapping.

Proof. Let $\eta \in L^Y$ be an H-compact set containing $f_L\left(1_X\right)$ and let $\rho \in {\Delta }^{\prime }$ .

Since $\eta$ is B-compact in $\left(L^Y,\Delta \right)$ which is fully stratified $L{T}_{2\frac{1}{2}}$ -space and

$L{R}_2$ -space, so $\eta \in {\Delta }^{\prime }$ and $\eta$ is H-bounded by Theorem 3.3 (ii). Thus $\eta \wedge \rho \in {\Delta }^{\prime }$ . Hence by Theorem 3.3 (iii), we have $\eta \wedge \rho \in L^Y$ is H-bounded. Thus $\eta \wedge \rho \in L^Y$ is closed and H-bounded. By HB-continuity of $f_L$ , then we have $f_L^{-1}\left(\eta \wedge \rho \right)\in {\tau }^{\prime }$ . But,

$f_L^{-1}\left(\eta \wedge \rho \right)=f_L^{-1}\left(\eta \right)\wedge f_L^{-1}\left(\rho \right)=f_L^{-1}\left(\rho \right)\wedge 1_X=f_L^{-1}\left(\rho \right)$ . So $f_L^{-1}\left(\rho \right)\in {\tau }^{\prime }$ . Hence $f_L$ is L-continuous mapping.

Theorem 5.2: If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is L-closed and L-almost continuous mapping, then $f_L^{-1}:\left(L^Y,\Delta \right)\to \left(L^X,\tau \right)$ is HB-continuous mapping.

Proof. Let $\eta \in L^X$ be an H-bounded and closed. Since $f_L$ is L-almost continuous mapping, then by Theorem 3.2 we have is H-bounded in $L^Y$ . Since $f_L$ is L-closed mapping, then $f_L\left(\eta \right)\in {\Delta }^{\prime }$ . Hence by Theorem 4.3, we have $f_L^{-1}$ is HB-continuous mapping.

Theorem 5.3: Let $\left(L^X,\tau \right)$ be an L-ts and $\left(L^Y,\Delta \right)$ be a fully stratified $L{T}_{2\frac{1}{2}}$

-space and $L{R}_2$ -space. If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is a bijective and L-almost continuous mapping, then $f_L^{-1}:\left(L^Y,\Delta \right)\to \left(L^X,\tau \right)$ is HB-continuous mapping.

Proof. Let $\eta \in L^X$ be an H-compact. Since $f_L$ is L-almost continuous mapping, then by Theorem 2.10, $f_L\left(\eta \right)$ is H-compact. Since $\left(L^Y,\Delta \right)$ is fully stratified $L{T}_{2\frac{1}{2}}$ -space and $L{R}_2$ -space, then $f_L\left(\eta \right)\in {\Delta }^{\prime }$ and $f_L\left(\eta \right)$ is H-bounded. Hence by Theorem 4.2, we have $f_L^{-1}$ is HB-continuous mapping.

Corollary 5.4: Let $\left(L^X,\tau \right)$ be an H-compact space and $\left(L^Y,\Delta \right)$ be a fully

stratified $L{T}_{2\frac{1}{2}}$ -space and $L{R}_2$ -space. If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is a bijective

and L-almost continuous mapping, then $f_L$ is a homeomorphism.

Proof. Follows from Theorem 5.1 and 5.3.

Theorem 5.5: Let $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ be a surjective L-mapping, then the following conditions are equivalent :

(i) $f_L$ is HB-continuous mapping.

(ii) For each $x_{\alpha }\in M\left(L^X\right)$ and each molecular net S in $L^X$ , $f_L\left(S\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ at $S\to x_{\alpha }$ .

(iii) $f_L\left(\lim \left(S\right)\right)\le HB.\lim \left(f_L\left(S\right)\right)$ for each S in $L^X$ .

Proof: (i) $\Rightarrow$ (ii): Let $x_{\alpha }\in M\left(L^X\right)$ and $S=\left\{S\left(n\right):n\in D\right\}$ be an molecular

net in $L^X$ which converges to $x_{\alpha }$ . Let $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ , by (i), we have

$f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ . Since $S\to x_{\alpha }$ then there is an $n\in D$ for all $m\in D$ , $m\ge n$

such that $S\left(m\right)\cancel{\le }{f}_L^{-1}\left(\eta \right)$ and so $f_L\left(S\left(m\right)\right)\cancel{\le }{f}_L{f}_L^{-1}\left(\eta \right)=\eta$ . Thus

$f_L\left(S\left(m\right)\right)\cancel{\le }\eta$ . Hence $f_L\left(S\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ .

(ii) $\Rightarrow$ (iii): Let S be a molecular net in $L^X$ and let $y_{\alpha }\in f_L\left(lim\left(S\right)\right)$ , then there exists $x_{\alpha }\in \lim \left(S\right)$ such that $y_{\alpha }=f_L\left(x_{\alpha }\right)$ . By (ii) we have

$f_L\left(x_{\alpha }\right)\in HB.\lim \left(f_L\left(S\right)\right)$ . Thus $f_L\left(\lim \left(S\right)\right)\le HB.\lim \left(f_L\left(S\right)\right)$ for each S in $L^X$ .

(iii) $\Rightarrow$ (i): Let $\eta \in L^Y$ be an HB-closed and $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in cl\left(f_L^{-1}\left(\eta \right)\right)$ . By Theorem 2.19, we have molecular net S in $f_L^{-1}\left(\eta \right)$ which

converges to $x_{\alpha }$ . Thus $x_{\alpha }\in \lim \left(S\right)$ and so $f_L\left(x_{\alpha }\right)\in f_L\left(\lim \left(S\right)\right)$ . By (iii),

$f_L\left(x_{\alpha }\right)\in f_L\left(\lim \left(S\right)\right)\le HB.\lim \left(f_L\left(S\right)\right)$ and so $f_L\left(S\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ . On the other hand, since S is molecular net in $f_L^{-1}\left(\eta \right)$ , then for each $n\in D$ , $S\left(n\right)\in f_L^{-1}\left(\eta \right)$ and so $f_L\left(S\left(n\right)\right)\le f_L\left(f_L^{-1}\left(\eta \right)\right)=\eta$ . Hence $f_L\left(S\left(n\right)\right)\le \eta$ for each $n\in D$ . Thus $f_L\left(S\right)$ is molecular net in $\eta$ . So we have $f_L\left(S\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ and $f_L\left(S\right)$ is molecular net in $\eta$ and so $f_L\left(x_{\alpha }\right)\in HB.cl\left(\eta \right)$ . But since $\eta$ is HB-closed L-subset, so $\eta =HB.cl\left(\eta \right)$ . Thus $f_L\left(x_{\alpha }\right)\in \eta$ . Hence $x_{\alpha }\in f_L^{-1}\left(\eta \right)$ . So $cl\left(f_L^{-1}\left(\eta \right)\right)\le f_L^{-1}\left(\eta \right)$ . Hence $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Then $f_L$ is HB-continuous mapping.

Theorem 5.6: If $f_L:\left(L^X,\tau \right)\to \left(L^Y,\Delta \right)$ is a surjective L-mapping. Then the following conditions are equivalent:

(i) $f_L$ is HB-continuous mapping.

(ii) For each $x_{\alpha }\in M\left(L^X\right)$ and each L-ideal I in $L^X$ , then $f_L\left(I\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ if $I\to x_{\alpha }$ .

(iii) $f_L\left(\lim \left(I\right)\right)\le HB.\lim \left(f_L\left(I\right)\right)$ for each I in $L^X$ .

Proof: (i) $\Rightarrow$ (ii): Let $x_{\alpha }\in M\left(L^X\right)$ and $I\to x_{\alpha }$ . Let $\eta \in HB{R}_{f_L\left(x_{\alpha }\right)}$ , by (i) , we have $f_L^{-1}\left(\eta \right)\in R_{x_{\alpha }}$ . Since $I\to x_{\alpha }$ then $f_L^{-1}\left(\eta \right)\in I$ . Since $x_{\alpha }\notin f_L^{-1}\left(\eta \right)$ , then $f_L\left(x_{\alpha }\right)\notin \eta$ , so $\eta \in f_L\left(I\right)$ . Hence $HB{R}_{f_L\left(x_{\alpha }\right)}\subseteq f_L\left(I\right)$ . Thus $f_L\left(I\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ .

(ii) $\Rightarrow$ (iii): Let I be an L-ideal in $L^X$ and let $y_{\alpha }\in f_L\left(\lim \left(I\right)\right)$ , then there exists $x_{\alpha }\in \lim \left(I\right)$ such that $y_{\alpha }=f_L\left(x_{\alpha }\right)$ . By (ii) we have $f_L\left(I\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ . So $y_{\alpha }=f_L\left(x_{\alpha }\right)\in HB.\lim \left(f_L\left(I\right)\right)$ . Hence $f_L\left(\lim \left(I\right)\right)\le HB.\lim \left(f_L\left(I\right)\right)$ for each I in $L^X$ .

(iii) $\Rightarrow$ (i): Let $\eta \in L^Y$ be an HB-closed set and $x_{\alpha }\in M\left(L^X\right)$ such that $x_{\alpha }\in cl\left(f_L^{-1}\left(\eta \right)\right)$ . By Theorem 2.23, there exists L-ideal I which converges to $x_{\alpha }$ such that $f_L^{-1}\left(\eta \right)\notin I$ . Moreover, $f_L\left(I\right)\le \left\{\rho \in L^Y:\eta \cancel{\le }\rho \right\}$ if $\lambda \in I$ with

$\eta \le \lambda$ , then there exists $\mu \in I$ satisfy $x_{\alpha }\notin \mu$ such that $f_L\left(x_{\alpha }\right)\notin \lambda$ . Since $\eta \le \lambda$ , then $f_L\left(x_{\alpha }\right)\notin \eta$ . This show that $x_{\alpha }\in \mu$ if $f_L\left(x_{\alpha }\right)\in \eta$ . Thus $f_L^{-1}\left(\eta \right)\le \mu$ . So $f_L^{-1}\left(\eta \right)\in I$ , a contradiction. Hence $\eta \notin f_L\left(I\right)$ . On the other

hand, by (iii), $f_L\left(x_{\alpha }\right)\in f_L\left(\lim \left(I\right)\right)\le HB.\lim \left(f_L\left(I\right)\right)$ . Thus $f_L\left(I\right)\stackrel{HB}{\to }{f}_L\left(x_{\alpha }\right)$ and so $f_L\left(x_{\alpha }\right)\in HB.cl\left(\eta \right)$ . But since $\eta$ is HB-closed L-subset, so $\eta =HB.cl\left(\eta \right)$ . Thus $f_L\left(x_{\alpha }\right)\in \eta$ . Hence $x_{\alpha }\in f_L^{-1}\left(\eta \right)$ . So

$cl\left(f_L^{-1}\left(\eta \right)\right)\le f_L^{-1}\left(\eta \right)$ . Hence $f_L^{-1}\left(\eta \right)\in {\tau }^{\prime }$ . Then $f_L$ is HB-continuous mapping.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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