Open Access Library Journal
Vol.06 No.05(2019), Article ID:92697,9 pages
10.4236/oalib.1105408
On Almost β-Topological Vector Spaces
Shallu Sharma, Sahil Billawria, Madhu Ram, Tsering Landol
Department of Mathematics, University of Jammu, Jammu and Kashmir, India
Copyright © 2019 by author(s) and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: April 16, 2019; Accepted: May 26, 2019; Published: May 29, 2019
ABSTRACT
In this paper, we have introduced a new generalized form of topological vector spaces, namely, almost β-topological vector spaces by using the concept of β-open sets. We have also presented some examples and counterexamples of almost β-topological vector spaces and determined its relationship with topological vector spaces. Some properties of β-topological vector spaces are also characterized.
Subject Areas:
Mathematical Analysis
Keywords:
β-Open Sets, δ-Open Sets, Regular-Open Sets, Almost β-Topological Vector Spaces
1. Introduction
The concept of topological vector spaces was introduced by Kolmogroff [1] in 1934. Its properties were further studied by different mathematicians. Due to its large number of exciting properties, it has been used in different advanced branches of mathematics like fixed point theory, operator theory, differential calculus etc. In 1963, N. Levine introduced the notion of semi-open sets and semi-continuity [2] . Nowadays there are several other weaker and stronger forms of open sets and continuities like pre-open sets [3] , precontinuous and weak precontinuous mappings [3] , β-open sets and β-continuous mappings [4] , δ-open sets [5] , etc. These weaker and stronger forms of open sets and continuities are used for extending the concept of topological vector spaces to several new notions like s-topological vector spaces [6] by M. Khan et al. in 2015, irresolute topological vector spaces [7] by M. Khan and M. Iqbal in 2016, β-topological vector spaces [8] by S. Sharma and M. Ram in 2018, almosts-topological vector spaces [9] by M. Ram et al. in 2018, etc. The aim of this paper is to introduce the class of almost β-topological vector spaces and present some examples of it. Further, some general properties of almost β-topological vector spaces are also investigated.
2. Preliminaries
Throughout this paper, (or simply X) and (or simply Y) mean topological spaces. For a subset , denotes the closure of A and denote the interior of A. The notation denotes the field of real numbers or complex numbers with usual topology and , represent the negligibly small positive numbers.
Definition 2.1 A subset A of a topological space X is said to be:
1) regular open if .
2) β-open [4] if .
Definition 2.2 A subset A of a topological space X is said to be δ-open [5] if for each , there exists a regular open set U in X such that .
The union of all β-open (resp. δ-open) sets in X that are contained in is called β-interior [10] (resp. δ-interior) of A and is denoted by (resp. ). A point x is called a β-interior point of if there exists a β-open V in X such that . The set of all β-interior points of A is equal to . It is well known fact that a subset is β-open (resp. δ-open) if and only if (resp. ). The complement of β-open (resp. δ-open, regular open) set is called β-closed (resp. δ-closed [5] , regular closed). The intersection of all β-closed (resp. δ-closed) sets in X containing a subset is called β-closure [10] (resp. δ-closure) of A and is denoted by (resp. ). It is also known that a subset A of X is β-closed (resp. δ-closed) if and only if (resp. ). A point if and only if for each β-open set V in X containing x. A point if for each open set O in X containing x.
The family of all β-open (resp. β-closed, regular open) sets in X is denoted by (resp. , ). If , , then (with respect to the product topology). The family of all β-open sets in X containing x is denoted by .
Definition 2.3 [11] A function from a topological space X to a topological space Y is called almost β-continuous at if for each open set O of Y containing , there exists such that .
Also we recall some definitions that will be used later.
Definition 2.4 [12] Let T be a vector space over the field . Let be a topology on T such that
1) For each and each open neighborhood O of in T, there exist open neighborhoods and of x and y respectively in T such that , and
2) For each , and each open neighborhood O of in T, there exists open neighborhoods of in and of x in T such that .
Then the pair is called topological vector space.
Definition 2.5 [8] Let T be a vector space over the field . Let be a topology on T such that
1) For each and each open neighborhood O of in T, there exist β-open sets and in T containing x and y respectively such that , and
2) For each , and each open neighborhood O of in T, there exist β-open sets containing in and containing x in T such that .
Then the pair is called β-topological vector space.
Definition 2.6 [13] Let T be a vector space over the field . Let be a topology on T such that
1) For each and each regular open set containing , there exist pre-open sets and in T containing x and y respectively such that , and
2) For each , and each regular open set containing , there exist pre-open sets in containing and containing x in T such that .
Then the pair is called an almost pretopological vector space.
Definition 2.7 [9] Let T be a vector space over the field . Let be a topology on T such that
1) For each and each regular open set containing , there exist semi-open sets and in T containing x and y respectively such that , and
2) For each , and each regular open set containing , there exist semi-open sets in containing and containing x in T such that .
Then the pair is called an almost s-topological vector space.
3. Almost β-Topological Vector Spaces
In this section, we define β-topological vector spaces and present some examples of it.
Definition 3.1 Let Z be a vector space over the field ( or with standard topology). Let be a topology on Z such that
1) For each and each regular open set containing , there exist β-open sets and in Z containing x and y respectively such that , and
2) For each , and each regular open set containing , there exist β-open sets in containing and containing x in Z such that .
Then the pair is called an almost β-topological vector space.
Some examples of almost β-topological vector space are given below:
Example 3.1 Let be the real vector space over the field , where with the standard topology and be the usual topology endowed on Z, that is, is generated by the base . Then is an almost β-topological vector space. For proving this, we have to verify the following two conditions:
1) Let . Consider any regular open set in Z containing . Then we can opt for β-open sets and in Z containing x and y respectively, such that
for each . Thus first condition of the definition of almost β-topological vector space is satisfied.
2) Let and . Consider a regular open set in containing . Then we have the following cases:
Case (I). If and , then . We can choose β-open sets in containing and in Z containing x, such that for each .
Case (II). If and , then . We can choose β-open sets in containing and in Z containing x, such that for each .
Case (III). If and (resp. and ), then . We can choose β-open sets in containing and in Z containing x, such that for each
(resp. ).
Case (IV). If and (resp. and ), then . We can select β-open neighborhoods (resp. ) in containing and (resp. in Z containing x, such that for each (resp. ).
Case (V). If and (resp. and ), then . We can select β-open neighborhoods (resp. ) in containing and (resp. in Z containing x, such that for each (resp. ).
Case (VI). If and , then . Then for β-open neighborhoods of in and of x in Z, we have for each .
This verifies the second condition of the definition of almost β-topological vector space.
Example 3.2 Let be the real vector space over the field with the topology generated by the base
, where denotes the set of irrational numbers. Then is an almost β-topological vector space.
Example 3.3 Consider the field with standard topology. Let be the real vector space over the field endowed with topology . Then is an almost β-topological vector space.
Example 3.4 Let be the topology induced by open intervals and the sets where with . Let be the real vector space over the field endowed with topology , where with the standard topology. Then is an almost β-topological vector space.
The above four examples are examples of almost β-topological vector spaces, we now present an example which don’t lie in the class of almost β-topological vector spaces.
Example 3.5 Let be the topology generated by the base and let this topology is imposed on the real vector space over the topological field with standard topology. Then fails to be an almost β-topological vector space. For, is regular open set in Z containing ( and ) but there do not exist β-open sets in containing −1 and in Z containing 0 such that .
Remark 3.1 By definitions, it is clear that, every topological vector space is an almost β-topological vector space. But converse need not be true in general. For, examples 3.2 and 3.3 are almost β-topological vector spaces which fails to be topological vector spaces.
Remark 3.2 The class of almost pretopological vector spaces and almost s-topological vector spaces lie completely inside the class of almost β-topological vector spaces.
4. Characterizations
Throughout this section, an almost β-topological vector space over the topological field will be simply written by Z and by a scalar, we mean an element from the topological field .
Theorem 4.1 Let A be any δ-open set in an almost β-topological vector space Z. Then , for each and each non-zero scalar .
Proof. Let . Then for some . Since A is δ-open, there exists a regular open set U in Z such that . . Since Z is an almost β-topological vector space, there exist β-open sets and in Z such that such that . Now . Since is β-open, . This shows that . Hence .
Further, let be arbitrary. Since A is δ-open, there exists a regular open set U in Z such that . Since Z is an almost β-topological vector space, there exist β-open sets in the topological field containing and in Z containing x such that . Now and hence . Thus is β-open in Z; i.e., .
Theorem 4.2 Let B be any δ-closed set in an almost β-topological vector space Z. Then for each and each non-zero scalar .
Proof. We need to show that . For, let be arbitrary and let W be any δ-open set in Z containing . By definition of δ-open sets, there is a regular open set U in Z such that . Then there exist β-open sets and in Z such that and . Since , then by definition, there is some
. Thus . Since B is δ-closed set, we have, . Therefore . Hence .
Next, we have to prove that . For, let be arbitrary and let W be any δ-open set in Z containing . By definition, there is a regular open set U in Z such that . Then there exist β-open sets containing in topological field and containing x in Z such that . Since , then there is some . Now . Thus . Therefore . Hence
.
Theorem 4.3 For any subset A of an almost β-topological vector space Z, the following assertions hold:
1) for each .
2) for each non zero scalar .
Proof. 1) Let . Then for some . Let O be an open set in Z containing z, then . Since Z is an almost β-topological vector space, then there exist containing x and y respectively such that . Since , then there is some . As a result, . Thus . Therefore
.
2) Let and let W be an open set in Z containing . Then , so there exist β-open sets containing in topological field and containing x in Z such that . Since , then there is some . Now
and hence
. Therefore .
Theorem 4.4 For any subset A of an almost β-topological vector space X, the following hold:
1) for each .
2) for each non-zero scalar .
Proof. 1) Let and let O be an open set in Z containing . Since Z is an almost β-topological vector space, there exist β-open sets and in Z such that , and . Since , there is some and hence
. Hence .
2) Let and O be an open set in Z containing . So there exist β-open sets in topological field containing and in Z containing x such that . As , and as a result, . Therefore . Hence .
Theorem 4.5 Let A be an open set in an almost β-topological vector space Z, then:
1) for each .
2) for each non zero scalar .
Proof. 1) Let and O be any open set in Z containing . Then there exist such that , and . Since , there is some . Now . Since A is open, . Thus ; that is, . Hence .
2) Let and O be any open set in Z containing . Then there exist β-open sets in topological field containing and in Z containing x such that . As , there is some . Thus . Since A is open, . Thus ; that is, . Hence .
Theorem 4.6 Let A and B be subsets of an almost β-topological vector space Z. Then .
Proof. Let and and let O be an open neighborhood of in Z. Since and is regular open, there exist such that , and
. Since and , there are
and . Then
. Thus ; that is, .
Theorem 4.7 For any subset A of an almost β-topological vector space Z, the following are true:
1) , and
2) , for each .
Proof. 1) We need to show that for each , . We know is δ-open. Then for each , there exists a regular open set U in Z such that . Since , for some . Since Z is almost β-topological vector space, then there exist β-open sets and in Z containing x and a respectively and . Thus . Since is β-open, then and therefore . Hence the assertion follows.
2) Let . Then there exists a regular open set U in Z such that . By definition of almost β-topological vector spaces, we have β-open sets and in Z containing -x and y respectively, such that . Thus . Hence .
Theorem 4.8 For any subset A of an almost β-topological vector space Z, the following are true:
1) , and
2) , for each non zero scalar .
Proof. Follows from the proof of above theorem by using second axiom of an almost β-topological vector space.
Theorem 4.9 Let Z be an almost β-topological vector space. Then
1) the translation mapping defined by , is almost β-continuous.
2) the multiplication mapping defined by , is almost β-continuous, where be non-zero scalar in .
Proof. 1) Let be an arbitrary. Let O be any open set in Z containing . As , we have Since Z is an almost β-topological vector space, there exist β-open sets and in Z containing x and y respectively such that . Thus . This proves that is almost β-continuous at y. Since was arbitrary, it follows that is almost β-continuous.
2) Let and O be any open set in Z containing . Then there exist β-open sets in the topological field containing and in Z containing x such that . Thus . This shows that is almost β-continuous at x and hence is almost β-continuous everywhere in Z.
Theorem 4.10 For an almost β-topological vector space Z, the mapping defined by , , is almost β-continuous.
Proof. Let and let U be regular open set in Z containing . Then, there exist β-open sets and in Z such that , and . Since is β-open in (with respect to product topology) such that and . It follows that is almost β-continuous at . Since is arbitrary, is almost β-continuous.
Theorem 4.11 For an almost β-topological vector space Z, the mapping defined by , , is almost β-continuous.
Proof. Follows from the proof of theorem 4.10 by using the second axiom of almost β-topological vector space.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Sharma, S., Billawria, S., Ram, M. and Landol, T. (2019) On Almost β-Topological Vector Spaces Open Access Library Journal, 6: e5408. https://doi.org/10.4236/oalib.1105408
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