^{1}

^{*}

^{1}

^{*}

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.

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 [

The axioms for a space to become a topological vector space or linear topological space have been given and studied by Kolmogroff [

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 [

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

Semi open sets in topological spaces were firstly appeared in 1963 in the paper of N. Levine [

Remember that, a set

If

Definition 1. Let

1)

2)

Recall that a topological vector space

real or complex numbers with their standard topologies) that is endowed with a topology such that:

1) Addition mapping

2) Multiplication mapping

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

2) for each

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.

Definition 2. A space

following two conditions are satisfied:

1) for each

2) for each

Remark 1. Topological vector spaces are independent of irresolute topological vector spaces.

The following example shows that

Example 1. Consider the vector space R_{(R)} endowed with the lower limit topology

the base

solute topological vector space.

Example 2. Let

The next example shows that

vector space.

Example 3. Consider the field

X be generated by the base

space, because for

Now, we show that

Case I: Let

Case II: Let

Now, we have to verify the second condition. For this we have four cases,

Case I: Let

Case II: Let

Case III: Let

Case IV: Let

Since, both conditions for irresolute topological vector spaces are satisfied, therefore,

lute topological vector space.

Theorem 1. Let

1) The (left) right translation

2) The translation

Proof. 1. Let W be a semi open neighbourhood of

2. Let

Remark 2. In topological vector spaces, every open set is translationally invariant whereas in irresolute topological vector spaces, every semi open set is translationally invariant.

Theorem 2. Let

1)

2)

Proof 1. Let

irresolute, by Theorem 1 , we have for any semi open neighbourhood A containing

semi open neighbourhood

2. Let

and

topological vector space and by Theorem 1(2),

Theorem 3. Let

X, then

Proof. Suppose

semi open sets is semi open, therefore

Corollary 1. Suppose

Theorem 4. Let

Proof. Let

Since

Theorem 5. Let

Proof. Let

V of y in X such that,

Let A be semi open in X. Then, by Theorem 3,

Definition 3. A mapping f form a topological space to itself is called irresolute-homeomorphism [

Theorem 6. Let

Proof. First, we show that

Similarly, we can prove that

Definition 4. An irresolute topological vector space

for each

Theorem 7. Every irresolute topological vector space is an irresolute homogenous space.

Proof. Let

Theorem 8. Suppose that

contains a non-empty semi open subset of X, then S is semi open in

Proof. Suppose U is a non-empty semi open subset in X, such that

is semi open subset of X for each

sets.

In general, intersection of two semi open sets is not semi open; however we have the following lemma.

Lemma 1. [

Lemma 2. [

Theorem 9. Every open subspace S of an irresolute topological vector space is also an irresolute topological vector space.

Proof. Suppose

satisfies the following properties.

1) For all

2) For any

Now, let

hood of

neighbourhoods

Again, for

As

X respectively. Since, S is open, therefore by Lemma 2, V is semi open in S. Hence for each semi open neighbourhood W of

Theorem 10. In irresolute topological vector spaces, for any semi open neighbourhood U of 0, there exists a semi open neighbourhood V of 0 such that

Proof. The proof is trivial, therefore omitted.

Theorem 11. Let A and B be subsets of an irresolute topological vector space. Then

Proof. Let

Theorem 12. Let

semi closed in X.

Proof. Let H be a semi open subspace of X. As right translation

therefore,

losed.

Theorem 13. Let

irresolute on X if it is irresolute at

Proof. Let

Theorem 14. Let

semi open, then for any set A, we have

Proof. As we know that

Theorem 15. Let

Proof. Let

Theorem 16. Let

set is semi-compact.

Proof. Let A be a semi-compact subsets of X. Let

zero

Definition 5. [

Theorem 17. Let

a semi-compact set A and semi-closed set B is semi-closed.

Proof. Let

open sets

of x. We claim that

Moizud Din Khan,Muhammad Asad Iqbal, (2016) On Irresolute Topological Vector Spaces. Advances in Pure Mathematics,06,105-112. doi: 10.4236/apm.2016.62009