Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsA view on Bb-Hausdorff M-spaces in Multiset topological spaces
Volume 3(4) · Pages 38–43
Research areasRough Sets and Fuzzy LogicFuzzy and Soft Set Theory
Abstract
Author-writtenThis paper deals with the concepts of B-open M-sets, Bb-closed M-sets and the Bb-continuity M-set function in M-topological spaces are discussed. Also, the concepts of Bb-Compact M-spaces, Bb-Hausdorff M-spaces, Bb-homeomorphism M-set functions and some applications of Bb-Hausdorff M-spaces on product topology are studied and some of their interesting relationships with respect to Bb-open M-sets are established.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- The study is based on the framework of M-topological spaces and open M-sets, which were previously introduced in the literature. View evidence ↗
- Hausdorff spaces are a fundamental concept in topology, originally defined by Felix Hausdorff in 1914. View evidence ↗
- The mathematical concept of compactness was first introduced by Maurice Fréchet in 1904. View evidence ↗
The question
- The authors aim to investigate B-open M-sets, Bb-closed M-sets, Bb-continuity, Bb-compactness, Bb-Hausdorff spaces, and Bb-homeomorphism functions within M-topological spaces. View evidence ↗
What they did
- The study defines X as a non-empty set and M as an element of [X]W, with a mapping CM from X to the set of whole numbers W. View evidence ↗
- The authors define a b-open M-set based on the union of the interior of the closure and the closure of the interior of the set. View evidence ↗
- A Bb-closed M-set is defined as a subM-set A where its B-closure is contained in any b-open M-set B that contains A. View evidence ↗
What they found
- The authors establish that every B-closed subM-set in an M-topological space is also a Bb-closed M-set. View evidence ↗
- The authors demonstrate that the converse of the relationship between B-closed and Bb-closed sets is not necessarily true. View evidence ↗
- It is proven that any Bb-closed subM-set of a Bb-compact M-space is itself Bb-compact. View evidence ↗
- The authors prove that in a Bb-Hausdorff M-space, every Bb-compact subM-set is Bb-closed. View evidence ↗
- The one-point Bb-compactification of an M-topological space is shown to be Bb-compact. View evidence ↗
- In a Bb-Hausdorff M-space, single points are proven to be Bb-closed. View evidence ↗
- An M-topological space is Bb-Hausdorff if and only if its diagonal is a Bb-closed M-set in the product space. View evidence ↗
- The product of two M-topological spaces is a Bb-Hausdorff M-space if and only if each individual space is Bb-Hausdorff. View evidence ↗
What it means
- The authors conclude that they have established significant relationships between Bb-Hausdorff M-spaces and Bb-open M-sets. View evidence ↗
Visual summary
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Product Space Bb-Hausdorff Property
AI-generated conceptual illustration (Abstract geometric representations of topological spaces). It explains the idea and does not show study data.
View source evidence ↗Introduction
The notion of M-topological space and the concept of open M-sets are introduced by Girish and sunil Jacob John [3] . Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. The term compact was introduced into mathematics by Maurice Fréchet in 1904. The main purpose of this paper is to study the concepts of B-open M-sets, Bb-closed M-sets and the Bb-continuity M-set function in M-topological spaces. Also, the concepts of Bb-Compact M-spaces, Bb-Hausdorff M-spaces, Bb-homeomorphism M-set functions and some applications of Bb-Hausdorff M-spaces on product topology are studied Throughout this paper X denote a non-empty set, M ∈ [X] W and C M : X ⟶ W where W is the set of all whole numbers.
1.1 Preliminaries Definition 1.1.1 [3] : Let M ∈ [X]w and τ ⊆ P * (M). Then τ is called a Multiset topology of M if τ satisfies the following properties.
1) The M-set M and the empty M-set ϕ are in τ.
2) The M-set union of the elements of any sub collection of τ is in τ.
3) The M-set intersection of the elements of any finite sub collection of τ is in τ. Definition 1.1.2 [3] : Given a subM-set A of an M-topological space M in [X]w, the interior of A is defined as the M-set union of all open M-sets contained in A and is denoted by Int(A). [3] : Given a subM-set A of an M-topological space M in [X]w, the closure of A is defined as the M-set intersection of all closed M-sets containing A and is denoted by Cl(A). i.e., Cl(A) = {K ⊆ M: K is a closed M-set and A ⊆ K} and [5] : Let (M, τ) be an M-topological space. If for every two M-singletons {k 1 /x 1 }, {k 2 /x 2 } ⊆ M such that x 1 ≠ x 2 , then there exist G, H ∈ τ such that {k 1 /x 1 } ⊆ G, {k 2 /x 2 } ⊆ H and G⋂H = ϕ . Hence (M, τ) is a Hausdorff M-space. i.e.,Bcl(A) = {K ⊆ M: each K is a B-closed M-set and A ⊆ K} with C Bcl(A) (x) = min {C K (x): A ⊆ K, each k ⊆ M is a Bclosed M-set}.
i.e., Int(
i.e., Int(
Definition 2.5: Let (M, T) be an M-topological space in [X] W and A be a subM-set of M. The M-set union of all B-open M-sets of (M, T) contained in A is called B-interior of A and is denoted by Bint(A).
i.e., Bint(A Proposition: Let (M, T) and (N, S) be any two M-topological spaces. For any M-set function f: (M, T) ⟶ (N, S), the following statements are equivalent:
(1) f is a Bb-continuous M-set function.
Proposition: Let (M, T), (N, S) and (P, R) be any three M-topological spaces. If a function f: (M, T) → (N, S) is a surjective Bbcontinuous M-set function and g: (N, S) → (P, R) is a continuous M-set function then g∘f: (M, T) → (P, R) is a Bb-continuous Mset function.
On Bb-Hausdorff M-Spaces
In this section, the concepts of Bb-Compact M-spaces, Bb-Hausdorff M-spaces and Bb-Homeomorphism M-set functions are studied and some applications of Bb-Hausdorff M-spaces on product topology are also discussed. ⊇ A, where I is an indexed set and max { C B i (x), i ∈ I is an indexed set} ≥ C A (x), for all x ∈ X, there exists a finite subset
Proposition: Let (M, T) be an M-topological space. Each Bb-closed subM-set of a Bb-compact M-space is Bb-compact.
Proof Let A be a Bb-closed subM-set of the Bb-compact M-space M and let
Then the M-sets A 1 ,….., A n cover f(M). Therefore f(M) = N is Bb-compact. Definition 3.5: Let (M, T) be an M-topological spaces and A be a subM-set of M. The collection T A = {N ∩ U: U is a open M-set} is an M-topology on A, called the subM-space M-topology on A. With the M-topology T A , the ordered pair (A, T A ) is called a sub M-space of M. Definition 3.6: Let (M, T) be an M-topological space and let m/x ∞ denote an ideal M-point, called the M-point at infinity, not
References
4 in the original paper- Chakrabarty, Kankana, Ioan Despi. n k -bags. Int. J Intell. Syst. 2007; 22(2):223-236.
- David Royster. Introduction to topology, 1999.
- Girish KP, Sunil Jacob John. On Multiset Topologies, Computer and Mathematics with Applications. 2012, 37-52.
- Mahanta J, Das D. Semi Compactness In Multiset Topology, Arvix: 1403.5642v2[math. GM] 21 Nov, 2014.