On Irresolute Topological Vector Spaces

In this paper, our focus is to investigate the notion of irresolute topological vector spaces. Irresolute topological vector spaces are defined by using semi open sets and irresolute mappings. The notion of irresolute topological vector spaces is analog to the notion of topological vector spaces, but mathematically it behaves differently. An example is given to show that an irresolute topological vector space is not a topological vector space. It is proved that: 1) Irresolute topological vector spaces possess open hereditary property; 2) A homomorphism of irresolute topological vector spaces is irresolute if and only if it is irresolute at identity element; 3) In irresolute topological vector spaces, the scalar multiple of semi compact set is semi compact; 4) In irresolute topological vector spaces, every semi open set is translationally invariant.


Introduction
If a set is endowed with algebraic and topological structures, then by means of a mathematical phenomenon, we can construct a new structure, on the bases of an old structure which is well known. This is the case we have introduced and discussed for beautiful interaction between linearity and topology in this paper. Although the new notion is similar to the notion of topological vector spaces, mathematically it behaves differently. To define irresolute topological vector space, we keep the algebraic and topological structures unaltered on a set but continuity conditions of vector addition and scalar multiplication are replaced by one of the characterizations of irresolute mappings.
A topological vector space [1] is a structure in topology in which a vector space X over a topological field F(R or C) is endowed with a topology τ such that the vector space operations are continuous with respect to τ .
The axioms for a space to become a topological vector space or linear topological space have been given and studied by Kolmogroff [2] in 1934 and von Neumann [3] in 1935. The relation between the axioms of topologi-cal vector space has been discussed by Wehausen [4] in 1938 and Hyers [5] in 1939. Also, Kelly [6] has done classical work on topological vector spaces. In the last decade, we can see the work of Chen [7], on fixed points of convex maps in topological vector spaces. Bosi et al. [8] and Clark [9] have researched on conics in topological vector spaces. More work, in recent years, has been done by Drewnowski [10], Alsulami and Khan [11] and Kocinac et al. [12]. In 2015, Moiz and Azam [13] defined and investigated s-topological vector spaces, which is a generalization of topological vector spaces.
The motivation behind the study of this paper is to investigate such structures in which the topology is endowed upon a vector space which fails to satisfy the continuity condition for vector addition and scalar multiplication or either. We are interested to study such structures for irresolute mappings in the sense of Levine. The concept of irresolute was introduced by Crossely and Hildebrand in 1972 as a consequence of the study of semi open sets and semi continuity in topological spaces, defined by Levine [14]. In this paper, several new facts concerning topologies of irresolute topological vector spaces are established.

Preliminaries
Throughout in this paper, X and Y are always representing topological spaces on which separation axioms are not considered until and unless stated. We will represent field by F and the set of all real numbers by  . δ and ∈ are assumed negligible small but positive real numbers.
Semi open sets in topological spaces were firstly appeared in 1963 in the paper of N. Levine [14]. A SO X ∈ , then ( ) where 1 X and 2 X are topological spaces and 1 2 X X × is a product space. It is worth mentioning that a set semi open in the product space cannot be expressed as product of semi open sets in the components spaces. Basic properties of semi open sets are given in [14], and of semi closed sets in [15] [16], and references therein. If ( ) F X is a vector space then e denotes its identity element, and for a fixed x X ∈ , and : x T X X → , y y x +  , denote the left and the right translation by x, respectively. The addition mapping , m x y x y = + , and the scalar multiplication mapping : be single valued function between topological spaces (continuity not assumed). Then: is termed as semi continuous [14], if and only if, is termed as irresolute [15], if, and only if, for each Recall that a topological vector space ( ) ( ) , F X τ is a vector space over a topological field F (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that: 1) Addition mapping ; , m x y x y x y X = + ∈ is continuous function.

2) Multiplication mapping
: ; , is continuous function (where the domains of these functions are endowed with product topologies).
Equivalently, we have a topological vector space X over a topological field F (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that: 1) for each , x y X ∈ , and for each open neighbourhood W of x y + in X, there exist neighbourhoods U and V of x and y respectively in X, such that U V W + ⊂ .
2) for each , Or equivalently, we have: topological Vector Space X over the field ( ) or F   with a topology on X such that ( ) , X + is a topological group and : M F X X × → is a continuous mapping.

Irresolute Topological Vector Spaces
In this section we will define and investigate basic properties of irresolute topological vector spaces. Examples are given to show that topological vector spaces are independent of irresolute topological vector spaces in general.
Topological vector spaces are independent of irresolute topological vector spaces. The following example shows that is neither a topological vector space nor an irresolute topological vector space.

Example 1.
Consider the vector space R (R) endowed with the lower limit topology τ on  , generated by the base is neither a topological vector space nor an irresolute topological vector space.

Example 2.
Let τ be a topology on X =  generated by the base is a topological vector space as well as irresolute topological vector space over the field  .
The next example shows that − ∈ ) containing λ and x in F and X respectively. Then, − ∈ ) containing λ and x in F and X respectively. Then, [ ) , V x x = + ∈ ) containing λ and x in F and X respectively. Then, Since, both conditions for irresolute topological vector spaces are satisfied, therefore, 1) The (left) right translation ( ) : M λ is an irresolute mapping.

( )
A x SO X + ∈ for every x X ∈ .

( )
A SO X λ ∈ for every x X ∈ . Proof 1. Let y X ∈ , and let z A y ∈ + , then we have to prove that z is a semi-interior point of A y + . Now, z x y = + , where x is some point in A. We can write ( ) x A y y A ∈ + + − = . By the right translation is an irresolute mapping.

Theorem 5. Let
Proof. First, we show that : is an irresolute homeomorphism. It is obviously bijective. By Theorem 1, y T is irresolute. Moreover This proves that, f is irresolute at x and hence on X.  ( ) . We know that  y x b a a x b a V b