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
denotes a completely distributive lattice with the smallest element 0 and the largest element 1 (
) and with an order reversing involution on it. An
is called a molecule of L if
and
implies
or
for all
. The set of all molecules of L is denoted by
. Let X be a nonempty set.
denotes the family of all mappings from X to L. The elements of
are called L-subsets on 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
by
and
, respectively. If
, then the constant mapping
is L-subset [6] . An L-point (or molecule on
), denoted by
,
is a L-subset which
is defined by
.
The family of all molecules
is denoted by
[7] . For
, we define
by the set
. An L-topology on X is a subfamily
of
closed under arbitrary unions and finite intersections. The pair
is called an L-topological space (or L-ts, for short) [8] . If
is an L-ts, then for each
,
,
and
will denote the closure, interior and complement of
. A mapping
is said to be an L-valued Zadeh function induced by a mapping
, iff
for every
and every
[7] . An L-ts
is called fully stratified if for each
,
[9] . If
is an L-ts, then the family of all crisp open sets in
is denoted by
i.e.,
is a crisp topological space [10] .
Definition 2.1 [11] : If
is L-ts, then
is called regular open set iff
. The family of all regular open sets is denoted by
. The complement of the regular open set is called the regular closed set and satisfy
. The family of all regular closed sets is denoted by
.
Definition 2.2 [11] : The L-valued Zadeh mapping
is called:
(i) Almost L-continuous iff
for each
.
(ii) Weakly L-continuous iff
for each
.
Definition 2.3 [12] : Let
be an L-valued Zadeh mapping and
, then
is defined as follows:
, for each
and call
the restriction of f on A. Where
denote the extension of
in
, that is for each
,
Definition 2.4 [13] : Let
be an L-ts and
. Then:
(i)
is called a remote neighborhood (R-nbd, for short) of
if
. The set of all R-nbds of
is called remoted neighborhood system and
is denoted by
.
(ii)
is called an
-remoted neighborhood (
-nbd, for short ) of
if there exists
such that
. The set of all
-nbds of
is
called
-remoted neighborhood system and is denoted by
.
Definition 2.5 [14] : Let
be an L-ts,
and
. Then
is called an:
(i)
-remoted neighborhood family of
, briefly
-RF of
, if for each
L-point
there is
such that
.
(ii)
-remoted neighborhood family of
, briefly
-RF of
, if there exists
such that
is an
-RF of
, where
, and
denotes the union of all the minimal sets relative to
.
Definition 2.6 [11] : Let
be an L-ts,
and
. Then
is called an:
(i) Almost
-
-remoted neighborhood family of
, (or briefly, almost
-
) of
, if for each L-point
there is
such that
.
(ii) Almost
-
-remoted neighborhood family of
, (or briefly almost
-
) of
, if there exists
such that
is an almost
-
of
.
Definition 2.7 [15] : Let
be an L-ts,
and
. Then
is called an
-regular closed remoted neighborhood family of
, briefly
-RCRF of
, if for each L-point
there is
such
that
.
Definition 2.8 [16] : Let
be an L-ts and
. Then
is called
-adherent point of
and write
iff
for
each
. If
, then
is called
-closed L-subset. The family
of all
-closed L-subset of X is denoted by
and its complement is called the family of all
-open L-subset and denoted by
.
Definition 2.9 [11] : Let
be an L-ts,
. Then
is called almost N-compact (or H-compact) set in
if for each
and every
-RF
of
there is
such that
is an almost
-
of
.
If
is H-compact set, then
is called H-compact space.
Theorem 2.10 [11] : Suppose that
is an L-almost continuous and
is an H-compact L-subset in
, then
is an H-compact L-subset in
.
Definition 2.11 [17] : An L-ts
is said to be:
(i)
-space iff for any
,
there is
such that
.
(ii)
-space iff for any
,
there is
,
such that
.
(iii)
-space iff for any
,
there is
,
such that
.
(iv)
-space (regular space) iff for all
,
and for each
there is
,
such that
and
.
(v)
-space iff it is
-space and
-space.
Theorem 2.12 [14] : Let
be an L-ts and every H-compact set in fully
stratified and
-space, then it is
-closed L-subset.
Theorem 2.13 [11] : An L-ts
is
-space iff for any
,
.
Proof. Let
be an
-space. For any
it is always true that
. Now, let
such that
and let
,
since
is
-space, there is
such that
. Now
implies that
for each
which implies that
which implies that
. Thus
. Hence
. Conversely, let
and
. Then
and so
. Hence there is
such that
.
Thus
is
-space.
Corollary 2.14 [11] : If
is
-space, then closed L-subset is
-closed L-subset and hence
is
-closed for any
.
Definition 2.15 [13] : Let
be a directed set. Then the mapping
and denoted by
is called a net of L-subsets in X. Specially, the mapping
is said to be a molecular net in
. If
and for each
,
then S is called a net in
.
Definition 2.16 [13] : Let
be an L-ts and
be a molecular net in
. S is called a molecular
-net (
), if for each
there exists
such that
whenever
, where
is the height of the molecular
.
Definition 2.17 [13] : Let
and
be a be molecular nets in
. Then T is said to be a molecular subnet of S if there is a mapping
that satisfies the following conditions:
(i)
(ii) For each
there is
such that
for each
,
.
Definition 2.18 [7] : Let
be an L-ts and S be a molecular net in
. Then
is called:
(i) a
-limit point of S, (or S
-converges to
) in symbols
if
for each
there is a
such for each
and
we have
. The union of all
-limit points of S are denoted by
.
(ii) a
-cluster (
-adherent) point of S, in symbols
if for each
and for each
there is a
such that
and
. The union of all
-cluster points of S is denoted by
.
Theorem 2.19 [13] : Let
be an L-ts,
and
. Then
iff there exists a molecular net S in
such that S is
-converges to
.
Theorem 2.20 [15] : Assume that
is a molecular net in an
L-ts
and
. Then
iff there exists a subnet T of S
such that
.
Theorem 2.21 [14] : Let
be an L-ts and
. Then
is H-compact set iff each
-net S contained in
has a
-cluster point in
with height
for any
.
Definition 2.22 [18] : The nonempty family
is called an ideal if the following conditions are satisfied, for each
(i)
(ii) If
and
, then
.
(iii) If
, then
.
Theorem 2.23 [19] : Let
be an L-ts,
and
. Then
iff there exists an ideal I in
such that I is
-converges to
and
.
Definition 2.24 [20] : An L-mapping
is called H-continuous if
for each
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
be an L-ts,
. Then
is called almost N-bounded (or H-bounded) set in
if for each
and every
-RF
of
, there is
such that
is an almost
-
of
.
If
is H-bounded set, then
is called H-bounded space.
Theorem 3.2: Suppose that
is an L-almost continuous and
is an H-bounded L-subset in
, then
is an H-bounded L-subset in
.
Proof. Let
be an H-bounded in
and let
be an
-RF of
(
), then
is an
-RCRF of
. We now will show that
is an
-RF of
. In fact,
since
is an L-almost continuous and
then
. According to the definition,
there exists
such that
, i.e.,
hence
for every
. This means that Q is an
-RF of
. Since
is an H-bounded set, there exists
such that
is an almost
-
of
. Thus for some
and for each
there exists
such that
. Since
is an L-almost continuous then it is L-weakly continuous and since
then
and so
. Consequently, there exists
and
satisfying
and
for each
. Thus,
is an almost
-
of
. By Definition 3.1, we have
an H-bounded L-subset in
.
Theorem 3.3: Let
be an L-ts and let
. Then the following statements are true:
(i) If
is H-compact set, then
is H-bounded set.
(ii) If
is H-bounded set and
, then
is H-bounded set.
(iii) If
is H-compact set and
, then
is H-bounded set.
Proof. (i) Let
be an H-compact set and let
be an
-RF of
and so
is
-RF of
. Since
is H-compact set, then there exists
such that
is an almost
-
of
. Thus
is H-bounded set.
(ii) Let
be an H-bounded set and
. let
be an
-RF of
. Since
is H-bounded set, then there exists
such that
is an almost
-
of
, thus there exists
such that
is an almost
-
of
. Hence
,
such that
. Since
, then
,
such that
. Hence
is an almost
-
of
and
so
is an almost
-
of
. Thus
is H-bounded set.
(iii) Let
be an H-compact set and
. let
be an
-RF of
and so
-RF of
. Since
is H-compact set, then there exists
such that
is an almost
-
of
, since
, then
is an almost
-
of
. Thus
is H-bounded set.
Theorem 3.4: Let
be an L-ts,
and
. Then
is H-bounded iff for each molecular
-net S contained in
has
-cluster point in
with height
.
Proof. Let
be an H-bounded set and
be an molecular
-net in
. If S does not have any
-cluster point in
with height
. Then for all
,
is not
-cluster point of S and so there exists
and
such that
for every
and
.
Put
, then
is an
-RF of
. According to
the hypothesis,
has a finite family
such that
is an almost
-
of
, that is for some
and each
there exists
(
) such that
. Put
, for each
, we have
, thus
. Since D is a directed set, then there is
such that
,
and
,
whenever
and so
.
This shows that for each
,
whenever
. This contradicts the hypothesis that S is a molecular
-net. Therefore, S has at least a
-cluster point in
with height
.
Conversely, assume that each molecular
-net S contained in
has an
-cluster point in
with height
and
is an
-RF of
. If for each
such that
is not almost
-
of
, that is, for each
there exists
there exists molecule
such that for each
,
. Put
and defined the order as follows:
iff
and
. Then
is an molecular
-net in
. Since
is an
-RF of
, then there exists
such that
and hence
. Because
. We take any
,
whenever
. Therefore
, which
contradicts to the hypothesis. Therefore there exists
such that
is almost
-
of
and hence
is H-bounded.
Theorem 3.5: If
fully stratified and
-space, then
is H-compact set iff
is
-closed and H-bounded set.
Proof. If
is H-compact set, then by Theorem 2.12 we have
is
-closed and by Theorem 3.3 (i) we have
is H-bounded. Conversely, let
be an
-closed and H-bounded set and let S be an
-net in
. Since
is H-bounded, then by Theorem 3.4 we have S has
-cluster point, say
in
with height
. By Theorem 2.20, then there is a subnet T of S such that T
-converges to
and so
by Theorem 2.19. Since
is
-closed, then
and so
, then by Theorem 2.21 we have
is H-compact set.
Theorem 3.6: If
is
-space, then
is H-bounded set iff
is H-bounded set.
Proof. If
is H-bounded set, then
is H-bounded set by Theorem
3.3 (ii). Conversely, suppose that
is H-bounded and
is an
-RF of
. Then for each
there is
such that
. Since
is
-space, then there is
there is
and there is
such that
and.
. Then the family
is an
-RF of
. Since
is H-bounded, then exists finite subset
of J such that
is an almost
-
of
. Since
,
, then
. Since
, then
is an almost
-
of
. Therefore
for
. Since
, and
is
-space, then by Theorem 2.13, we have
and so
is an almost
-
of
and since
, then
is an almost
-
of
. Hence
is H-bounded set.
Theorem 3.7: If
is
-space, then
is H-bounded set iff
is L-subset of H-compact set.
Proof. If
is H-bounded, then by Theorem 3.6 and corollary 2.14, we have
is
-closed and H-bounded set, hence by Theorem 3.5, we have
is H-compact set. Conversely, If
is L-subset of H-compact set, then by Theorem 3.3 (iii), we have
is H-bounded set.
Definition 3.8: Let
be an L-ts and
. If
is closed and H-bounded set, then
is called HB-remoted neighborhood of
(HBR-nbd, for short) of
if
. The set of all HBR-nbds of
is denoted by
We note that
,
The following example shows that the converse is not true in general
Example 3.9: Let
,
, and let
. Then
is L-ts. We have
. Now, we show that
is not H-bounded set.
Let
, then
is .8-RF of
. But for each
, any finite subfamily
is not almost
-
of
. Thus
is not almost
-
of
. Thus
is not H-bounded set
and so
. Hence
.
Definition 3.10: Let
be an L-ts and
. Then
is called an H-bounded adherent point of
and write
iff
for each
. If
, then
is called HB-closed
L-subset. The family of all HB-closed L-subsets is denoted by
and its complement is called the family of all HB-open L-subsets and denoted by
.
Theorem 3.11: Let
be an L-ts and let
. Then the following statements are true:
(i)
.
(ii) If
and
then
.
(iii)
.
(iv)
.
Proof. (i) Let
such that
, then there exists
such that
. Since
and so
and hence
. Thus
.
(ii) Let
such that
, then there exists
such that
. Since
, then
and so
. Thus
.
(iii) Suppose
such that
. According to
Definition 3.10, we have
for each
. Hence, there exists
such that
with
and so
, that is,
. This shows that
. On the other hand,
follows from (i) and so
. Therefore,
.
(iv) On account of (i) and (iii).
is an HB-closed set containing
,
and so
. Conversely, in case
sand
, then
for each
. Hence, if
is an HB-closed set containing
, then
, and then
.
This implies that
. Hence
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
be an L-subset, where
, then
is closed L-subset because
. But
is not H-bounded set.
Theorem 3.13: Let
be an L-ts. The following statements hold:
(i)
.
(ii) If
, then
.
(iii) If
, then
.
(iv) Every H-bounded and closed set is HB-closed.
(v)
is HB-closed iff there exists
such that
for
each
with
Proof. (i) Obvious.
(ii) Let
and
such that
, then for each
we have
and so
for some
. Hence
for some
. Since
is HB-closed set, then
for some
and so
for some
and hence
. Thus
( ∗ )
Conversely, since
then
(
). Hence from (
) and (
) we have
. Thus
.
(iii) Let
and
such that
, then for each
we have
and so
for each
. Hence
for each
. Since
is HB-closed set, then
for each
and so
for each
and
hence
. Thus
(
).
Conversely, since
then
(
). Hence from (
) and (
) we have
. Thus
.
(iv) Let
be an H-bounded and closed set and let
such
that
, since
is H-bounded and closed set, then
, since
then
and so
. Therefore
is HB-closed set.
(v) Suppose that
is HB-closed set,
and
. By Definition 3.9, there exists
with
. Conversely, provided that the condition is satisfied. If
is not HB-closed set, then there exists
such that
and
. Hence
for each
. It
conflicts with the hypothesis, and so
is HB-closed set.
Theorem 3.14: Let
be an L-ts and
. Then
iff
for each
.
Proof. It follows directly from Theorem 3.13 (v).
Theorem 3.15: Let
be an L-ts and
. Then the mapping
is called closure operator of HB-boundedness iff it satisfies:
(i)
.
(ii)
.
(iii)
.
(iv)
.
A closure operator of HB-boundedness
generates L-topology
on
as:
.
Proof. It follows directly from Theorems 3.11 and 3.13.
Theorem 3.16: Let
be an L-ts. Then:
(i)
.
(ii) If
is H-bounded space, then
.
Proof. (i) Let
, then
. Since
, hence
and so
.
(ii) We note that
from (i). Now, let
then
. Since
is H-bounded and
, then
is H-bounded (By Theorem 3.3 (ii)) and by Theorem 3.13 (iv) we have
is HB-closed set and so
. Thus
.
Definition 3.17. Let
be an L-ts,
and
. We say that
is the HB-interior of
.
The following Theorem shows the relationships between HB-closure operator and HB-interior operator.
Theorem 3.18: Let
be an L-ts and
. Then the following are true:
(i)
is HB-open iff
.
(ii)
and
.
(iii)
and
.
(iv)
.
(v) If
and
then
.
(vi)
.
Proof. (i) Let
be an HB-open set, then
and so
.
Conversely, let
, since
. Therefore
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
be an L-ts. The following statements hold::
(i)
.
(ii) If
, then
.
(iii) If
, then
.
Definition 3.20: Let
be an L-ts and S be a molecular net in
. Then
is called
(i) limit point of S [13] , (or S converges to
) in symbol
if for
every
there is
such for each
and
we have
. The union of all limit points of S is denoted by
.
(ii) H-bounded limit point of S, (or S HB-converges to
) in symbol
if for every
there is an
such that
and
, we have
. The union of all HB-limit points of S is denoted by
.
Theorem 3.21: Suppose that S is a molecular net in
,
and
. Then the following statements hold:
(i) If
, then
.
(ii)
iff
.
(iii)
.
(iv)
(resp.
), iff there exists a molecular net S in
such that S is HB-converges (resp. converges) to
.
(v)
is HB-closed set in
.
Proof. (i) Let
and let
. Since
, then
Since
, then for every
there is
such for each
and
, we have
. Thus
.
(ii) Let
and let
. Since
, then
. Therefore there exists
such that
and
. Then
and so there is
much
for each
and
we have
, but since
so
. Conversely, let
, then by Definition 3.20 (ii) we have
(iii) Let
and let
. Since
, then
. And since
, then for each
there is
such for each
and
, we have
and so
. Hence
. So
.
(iv) Let
such that
, then
for each
. Since
, then there exists
such that
with
. Since the pair
is a directed set and so we can define a molecular net
as follows
for each
Hence S is a molecular net in
. Now let
such that
, so we have there exists
and so
. Hence S is HB-converges to
.
Conversely, let S be a molecular net in
such that S is HB-converges to
then for each
there is
such for each
and
,
we have
. Since
for each
,
. So
and
hence
for each
. This means that
.
(v) Let
, then
for each
and then there exists
such that
and
. Then for each
, there is
much for each
and
we have
and so
. Hence
. Thus
and so
is HB-closed set.
Definition 3.22: Let
be an L-ts and I be an ideal in
. Then
is called:
(i) limit point of I [18] , (or I converges to
) in symbol
if
. The union of all limit points of I is denoted by
.
(ii) H-bounded limit point of I, (or I HB-converges to
) in symbol
if
. The union of all HB-limit points of I is denoted by
.
Theorem 3.23: Suppose that I is an ideal in
,
and
. Then the following statements hold:
(i) If
, then
.
(ii)
iff
.
(iii)
.
(iv)
iff there exists an ideal I in
such that
and
(v)
is HB-closed set in
.
Proof. (i) Let
then
. Since
, then
. Thus
.
(ii) Let
and let
. Since
and
, then
. Therefore there exists
such that
and
. Then
and so
hence
. Thus
. Conversely, let
, then by Definition 3.22 (ii) we have
.
(iii) Let
and let
. since
, so for each
,
and since
so
. Hence
. So
.
(iv) Let
such that
. The family
is an ideal in
. Now we show that
. Since
, then for each
,
. So By definition of I we have
. Finally, we show that
. Let
, since
, then
. So
. Thus
.
Conversely, let I be an ideal in
such that
and
. Then for each
,
. Since
,
, then
and so
.
(v) Let
, then
for each
and then there exists
such that
and
. Since
and
then
for each
. Since
then
. But
and so
. Thus
and so
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
is called :
(i) HB-continuous at
if
for each
(ii) HB-continuous if
for each
is closed and H-bounded.
Theorem 4.2: Let
be an L-continuous mapping. Then the following properties are equivalent :
(i)
is HB-continuous.
(ii)
is HB-continuous at
for each
.
(iii) If
and
is H-bounded, then
.
(iv) If
is H-bounded, then
.
Proof. (i)
(ii): Let
be an HB-continuous and
,
then
. Since
, then
And so
. Thus
is HB-continuous at
for each
.
(ii)
(i): Let
be an HB-continuous at
for each
. If
is not HB-continuous, then there is
is H-bounded and closed such that
, i.e.,
. Then there exists
such that
and
implies that
, since
is
closed and H-bounded, then
. But
, this contradiction. Thus
is HB-continuous mapping.
(i)
(iii): Let
be an HB-continuous and
such that
is H-bounded and so
is H-bounded and closed. By (i), we have
. Since
, then
.
(iii)
(i): Let
be an H-bounded and closed, then
. By (iii), we have
, thus
, then
. Hence
is HB-continuous mapping.
(iv)
(iii): Let
and
be an H-bounded. By (iv), we have
. Thus
.
(iv)
(ii): Let
and
. Then
is closed and H-bounded set,
and so
. By (iv), we have
and
hence
. Thus
is HB-continuous mapping at
for each
.
(iv)
(i): Let
be a closed and H-bounded set. By (iv), we have
. Thus
is HB-continuous mapping.
Theorem 4.3: Let
be an L-surjective mapping. Then the following conditions are equivalent:
(i)
is HB-continuous mapping.
(ii) For each
,
,
(iii) For each
,
,
(iv) For each
,
,
(v) For each HB-open L-subset
in
, then
is open L-subset in
,
(vi) For each HB-closed L-subset
in
, then
is closed L-subset in
.
Proof. (i)
(ii): Let
and
such that
. Then
. Let
. So by (i) and by Theorem 4.3, we have
. Since
, then
. Since
is L-surjective then
and
so
. Hence
.
(ii)
(iii): Let
. Then
. By (ii) we have
. So
. Thus
. Since
, then
.
(iii)
(iv): Let
. By (iii), we have
Since
and
. So
. Thus
.
(iv)
(v): Let
be an HB-open L-subset in
. Then
and by (iv), we have
, so
. Thus
.
(v)
(vi): Let
be an HB-closed L-subset in
. By (v), we have
. Then
and so
.
(vi)
(i): Let
be an closed and H-bounded set, then
is HB-closed L-subset in
. By (vi), we have
. Thus
is HB-continuous mapping.
Theorem 4.4: If
is HB-continuous mapping, then
is HB-continuous mapping.
Proof. Let
such that
is H-bounded set, then
is
H-bounded and closed in
. Therefore
and
is H-bounded in
. Since
is HB-continuous mapping, the by Theorem 4.2 (iii), we have
, thus
.
Hence
consequently,
is
HB-continuous mapping.
Theorem 4.5: If
is HB-continuous mapping and
then
is HB-continuous mapping.
Proof. Let
be an H-bounded and closed set. Since
is HB-continuous mapping, then
and since
. Hence
is HB-continuous mapping.
Theorem 4.6: Every
L-continuous mapping is HB-continuous mapping.
Proof. Let
be an L-continuous and let
be an closed and H-bounded set, then
. Thus
is HB-continuous mapping.
The following example shows that the converse is not true in general.
Example 4.7: Let
be the usual interval base of the relative L-topology on
induced by the set of real numbers. Define a L-topology
on
generated by the base consisting of,
,
and
where
Let
be the L-topology on I such that the complements of any number of
is countable L-subset in I (i.e., the support of the L-subset is countable). Let
be a function defined by
, for all
. Then it can be see that
is HB-continuous but not L-continuous mapping.
Theorem 4.8: A mapping
is L-continuous mapping iff it is HB-continuous mapping.
Proof. Since
, then necessity is evident. Now, we suppose that
is HB-continuous and
. Then by Theorem 4.3 (iii) we have
and so
. Thus
is L-continuous mapping.
Theorem 4.9: Let
be an L-mapping and
is H-bounded space. Then
is L-continuous mapping iff
is HB-continuous mapping.
Proof. By Theorem 4.6 we need only to investigate the sufficiency. Let
. Since
is H-bounded space then by Theorem 3.2(ii), we have
is H-bounded set and so
is HB-closed L-subset. By HB-continuity of
, we have
. Hence
is L-continuous mapping.
Theorem 4.10: If
is HB-continuous, then
is H-continuous mapping.
Proof. Follows from the fact that every H-compact set is H-bounded set.
Theorem 4.11: Let
be an L-mapping and
be
-space. Then
is H-continuous iff
is HB-continuous mapping.
Proof. Let
be an HB-continuous mapping and let
be a closed and H-compact, then by Theorem 3.3 (i), we have
is H-bounded and closed. Since
is HB-continuous then
. Thus
is H-continuous.
Conversely, let
be an H-continuous and let
be a closed and H-bounded. Then
is H-compact and closed. Since
is H-continuous, then
. Thus
is HB-continuous mapping.
Remark 4.12: For an L-mapping
, we obtain the following implications:
L-continuity HB-continuity H-continuity.
None of these implications are reversible. However, if it
is H-bounded (resp.
-) 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
is L-continuous and
is HB-continuous, then
is HB-continuous.
Proof. Let
be a closed and almost N-compact. Since
is HB-continuous, then
and since
is L-continuous, then
Hence
HB-continuous mapping.
Theorem 4.14: If
and
are L-ts's and
such
that
and
is L-mapping and
are
HB-continuous mappings, then
is HB-continuous mapping.
Proof. Let
be an N-almost bounded and closed then
Hence
. Thus
is HB-continuous mapping.
Theorem 4.15: If
is HB-continuous mapping, injective,
is
-space and H-bounded, then
is
-space.
Proof. Let
such that
. Since
is injective L-mapping, then
and
. Since
is
-space, then
are closed L-subsets in
. Since
is H-bounded, then
are H-bounded L-subsets. Since
is HB-continuous mapping, then
and
are closed L-subsets in
. Hence
is
-space.
5. Characterizations of HB-Continuous Mappings in L-Topological Space
Theorem 5.1: Let
be an HB-continuous mapping and
be a fully stratified
-space and
-space. If
is contained in
some H-compact set of
, then
is L-continuous mapping.
Proof. Let
be an H-compact set containing
and let
.
Since
is B-compact in
which is fully stratified
-space and
-space, so
and
is H-bounded by Theorem 3.3 (ii). Thus
. Hence by Theorem 3.3 (iii), we have
is H-bounded. Thus
is closed and H-bounded. By HB-continuity of
, then we have
. But,
. So
. Hence
is L-continuous mapping.
Theorem 5.2: If
is L-closed and L-almost continuous mapping, then
is HB-continuous mapping.
Proof. Let
be an H-bounded and closed. Since
is L-almost continuous mapping, then by Theorem 3.2 we have is H-bounded in
. Since
is L-closed mapping, then
. Hence by Theorem 4.3, we have
is HB-continuous mapping.
Theorem 5.3: Let
be an L-ts and
be a fully stratified
-space and
-space. If
is a bijective and L-almost continuous mapping, then
is HB-continuous mapping.
Proof. Let
be an H-compact. Since
is L-almost continuous mapping, then by Theorem 2.10,
is H-compact. Since
is fully stratified
-space and
-space, then
and
is H-bounded. Hence by Theorem 4.2, we have
is HB-continuous mapping.
Corollary 5.4: Let
be an H-compact space and
be a fully
stratified
-space and
-space. If
is a bijective
and L-almost continuous mapping, then
is a homeomorphism.
Proof. Follows from Theorem 5.1 and 5.3.
Theorem 5.5: Let
be a surjective L-mapping, then the following conditions are equivalent :
(i)
is HB-continuous mapping.
(ii) For each
and each molecular net S in
,
at
.
(iii)
for each S in
.
Proof: (i)
(ii): Let
and
be an molecular
net in
which converges to
. Let
, by (i), we have
. Since
then there is an
for all
,
such that
and so
. Thus
. Hence
.
(ii)
(iii): Let S be a molecular net in
and let
, then there exists
such that
. By (ii) we have
. Thus
for each S in
.
(iii)
(i): Let
be an HB-closed and
such that
. By Theorem 2.19, we have molecular net S in
which
converges to
. Thus
and so
. By (iii),
and so
. On the other hand, since S is molecular net in
, then for each
,
and so
. Hence
for each
. Thus
is molecular net in
. So we have
and
is molecular net in
and so
. But since
is HB-closed L-subset, so
. Thus
. Hence
. So
. Hence
. Then
is HB-continuous mapping.
Theorem 5.6: If
is a surjective L-mapping. Then the following conditions are equivalent:
(i)
is HB-continuous mapping.
(ii) For each
and each L-ideal I in
, then
if
.
(iii)
for each I in
.
Proof: (i)
(ii): Let
and
. Let
, by (i) , we have
. Since
then
. Since
, then
, so
. Hence
. Thus
.
(ii)
(iii): Let I be an L-ideal in
and let
, then there exists
such that
. By (ii) we have
. So
. Hence
for each I in
.
(iii)
(i): Let
be an HB-closed set and
such that
. By Theorem 2.23, there exists L-ideal I which converges to
such that
. Moreover,
if
with
, then there exists
satisfy
such that
. Since
, then
. This show that
if
. Thus
. So
, a contradiction. Hence
. On the other
hand, by (iii),
. Thus
and so
. But since
is HB-closed L-subset, so
. Thus
. Hence
. So
. Hence
. Then
is HB-continuous mapping.