Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 259 https://internationalpubls.com Compactness and Connectedness in Beta Weakly Semi – Closed Sets in Topological Spaces S. Saranya*1 , V.E. Sasikala*2 *1 Research Scholar, *2 Assistant Professor, Research Supervisor Corresponding Author, Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies, (VISTAS), Pallavaram, Chennai, India. Corresponding author mail id.sasikala.sbs@velsuniv.ac.in Article History: Received: 10-05-2024 Revised: 25-06-2024 Accepted: 07-07-2024 Abstract: This research presents an innovative class of Beta weakly semi-CS, namely Compactness and Connectedness in Beta weakly semi-CS in TS. Throughout this paper, ws- Compactness and ws-Connectedness were examined to get the fundamental facts in the Beta weakly semi-CS. In this paper, the notion of countable βws- compact in TS were explored and ws – Connectedness (Cws) in TS were also studied to get results. The ws - Cws and ws – Compactness fulfilled most of the connectedness and compactness properties in TS. Here, many characterizations were obtained along with some of their features. The paper concludes on how it relates to other kinds of functions and beta ws-Compactness in TS and its characteristics were studied to obtain results theoretically. Keywords: Beta weakly semi – closed sets (ws - closed), Beta weakly semi – open sets (ws-open), Beta weakly semi-closed sets – compactness (ws – compactness), Beta weakly semi closed sets – connectedness (ws – connectedness). 1. Introduction: The fundamental concepts of connectedness and compactness play crucial roles in general topology and various other branches of mathematics. Numerous researchers have investigated into exploring their fundamental properties, leading to inspirations for generalizing these concepts to innovative extents. [1], “ On b-open sets, Math Vesnik in topological spaces’’. K. Rekha [2], “**b-Compactness and **b-Connectedness in Topological Spaces’’. [3], “ θ-b-continuous functions, Acta Math. Hungar’’. D. Sivaraj and V.E. Sasikala [4-6], “A Study on Soft α−open sets” in topological spaces. “On soft semi weakly generalized closed set’’ and “Soft swg Separation Axioms in soft topological spaces’’. V. Kavitha, V. E. Sasikala [7], “Beta Generalized CS’’ in Topological Spaces. [8], “Minimal weakly open sets and maximal weakly closed sets in topological spaces”. [9], “Some properties of contra-γ-continuous functions’’. [10], “On generalized continuous maps’’ in topological spaces.[11], Studies “On Generalizations of Closed Maps and Homeomorphisms in Topological Spaces”.[12],“On gpr-continuous functions’’ in topological spaces. A. Pushpalatha,[13] Studies “On Generalizations of Mapping in Topological Spaces’’. [14], “A Study on Generalizations of Closed Sets and Continuous Maps’’ in Topological and Bitopological Spaces. This research aims to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 260 https://internationalpubls.com introduce the ideas extending βws - compactness and βws – connectedness (Cws) within TS, along with providing characterizations for these concepts. Review of Literature: A thorough review of theoretical contexts and terminology that have contributed to the current knowledge of the Beta weakly semi-closed sets in topological spaces were studied. Whenever A is g*s-compact of the subspace of X is called g*s-compact [15]. A topological space 𝑋 is said to be generalized semi-open connected (briefly 𝑔𝑠𝑜-connected) if 𝑋 cannot be written as the union of two non-empty disjoint 𝑔𝑠𝑜-open sets [16]. A space X is said to be **b-compact if every “**b-open cover {𝐴 } of X contains a finite sub collection that also covers X’’ [1-3]. The methods applied for studying these sets, the extensiveness of the research, and the way these concepts are used to better comprehend authentic topological issues. The emergence of beta weakly semi-closed sets in topological spaces, which provide an innovative viewpoint, is anticipated to advance our understanding of the field of topological spaces and its basic features. 2. Preliminary Notes: All through this study, ( X, τ ) and ( Y, σ ) as generic topological spaces were considered unless specified otherwise, without assuming any separation axioms. For the closure of set A is X, we denote it as cl (A), and Int (A) represents the interior of A. Definition:2.1.[1] “A subset A of X is said to be b-open [1] if A⊆Int(cl(A))∪cl(Int(A)). The complement of b-open set is said to be b-closed. The family of all b-open sets (respectively b-closed sets) of (X, τ) is denoted by bO(X, τ) [respectively bcl(X, τ)]”. Definition 2.2.[1] Let A  X. Then (i) “ b-interior [1] of A is the union of all b-open sets contained in A”. (ii)“b-closure [1] of A is the intersection of all b-closed sets containing A. The b-interior [respectively b-closure] of A is denoted by b-Int(A) [respectively b-Cl(A)]”. 3. ws-Compactness in TS: This section examines the idea of βws-Compactness other than subsequently explores it’s characteristics and also provides its fundamental properties Definition 3.1. The collection {Ģi :iϵI} of ws-O-S in a TS X is known as ws-O-C  Ģ related to X if Ģ⋃ iϵI Ģi Definition 3.2. The TS X were named as ws-compact with each and every one ws-O-C of X include a finite sub cover. Definition 3.3. A  Ģ of the TS X were named as ws-compact related to X,  collection {Ģ i :iϵI} of ws – O  X  Ģ  ⋃ Ai iϵI  finite  I0 of I  Ģ  ⋃ Ai iϵI . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 261 https://internationalpubls.com Definition 3.4. A  Ģ of a TS X were named as ws - Compact if Ģ is ws-Compact  X. Theorem 3.5. A ws-C  ws -compact space were considered as ws-compact relative to X. Proof. Assume that Ģ be a ws-C  TS in X. Thence Ģc is ws-O in X, 𝒮 = {Ģi :iϵI} exist a ws-O-C of Ģ by ws-O  X. After that, 𝒮* = 𝒮 ∪ Ģc is a ws-O-C of X. That is X = [∪{Ģi :iϵI}]∪ Ģc. By the assumption that, X will be considered as ws-compact, ∴, 𝒮* reduction to a finite subcover of X for example, X = Ģi1∪ Ģi2∪…∪ Ģin ∪ Ģc, Ģik 𝒮*. Since Ģ and Ģc are disjoint. ∴, Ģ  Ģi1∪ Ģi2∪…..∪ Ģin𝒮. Thus a ws-O-C 𝒮 of Ģ  finite subcover, proving that Ģ is the βws - Compact related to X. Theorem 3.6: Each ws-compact space are considered as compact. Proof. Assume that X are considered as ws-Compact space.{Ģi :iϵI} is an O-C of X. Thenceforth {Ģi :iϵI} will be a ws-O-C of X Since every O-S is also a βws – O-S, ∴, X is ws-compact, the ws-O-C {Ģi :iϵI} of X includes a finite subcover talks {Ģi , i ϵ n} within X. ∴, X is compact. Theorem 3.7: Each and all GO-Compact space are considered as ws-Compact. Proof. Assuming X relate as a GO-Compact space. Assuming, {Ģi :iϵI} relate as a ws-O-C of X by ws- O-S in X. Afterward, every ws-O-S is g-O,∴, {Ģi :iϵI} is g-O-C regarding X. ∴, X is GO- Compact, the g-O-C {Ģi :iϵI} X has a finite sub cover say {Ģi i ϵ n} of X . ∴, X is ws - Compact. 4. Countable ws-Compactness in TS: In this section, we explored that the notion of countable βws-compact in TS besides that investigate most of its characteristics. Definition 4.1. A TS X is named as countable ws-Compact if each and all countable ws-O-C of` X includes a finite sub cover. Theorem 4.2. Suppose X considered as countable ws-Compact space, thenceforth ⴇ would be considered as countable compact. Proof. Consider, {Ģi :iϵI} be present countable O-C of X along with O-S in X. Thenceforth {Ģi :iϵI} is countable ws-O-C regarding X. Given that X would be countable ws-Compact, that countable Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 262 https://internationalpubls.com ws-O-C of X includes a finite subcover, for example {Ģi .i ϵn}. ∴, X are considered as countable Compact. Theorem 4.3. Each and every ws-compact space is considered to be Countable ws-compact. Proof. Suppose that Ҳ will be considered as ws-Compact space, {Ģi :iϵI} will be considered as countable ws-O-C of Ҳ including ws-O-S. Following that {Ģi :iϵI} is a ws-O-C {Ģi :iϵI} of X whose finite sub cover say {Ģi, iϵn}. ∴, X is countable ws-Compact. Theorem 4.4. The countable ws-Compact space under ws-irresolute function are considered as countable ws- Compact. Proof. Suppose ⴇ : X → Y have a ws-irresolute function after a countable ws-Compact space Ҳ continuously TS Y, {Ģi : iϵI} will be considered as countable ws-O-C of Y. Subsequently, {ⴇ - 1(Ģi): iϵI} are considered as countable ws-O-S of X such as ⴇ remains ws-irresolute. when X is countable ws-Compact, the countable ws-O-C {ⴇ -1(Ģi) : iϵI} of X include a finite subcover say {ⴇ -1(Ģi) : in}. ∴, X = ∪i∈I ⴇ -1 (Ģi) ⴇ (X) = ∪i∈I Ģi. Then Y = ∪i∈I Ģi {Ģ1, Ģ2, Ģ3 ,…, Ģn} is a finite  { Ģi : iϵI} used for Y. ∴, Y is countable ws - Compact. Theorem 4.5: A space X is countable ws - Compact if all countable unit of ws-CS that is X having finite intersection property has a non-empty intersection. Proof: The X is considered as countable ws-Compact, {Fi:iϵI} is included to countable unit of ws-CS along through finite ∩ property. To show that ∩i∈I n (Fi)  . To take ∩i∈I n (Fi) = , Then X ∪i∈I Fi = X  ∪i∈I (X - Fi) = X. The {X - Fi : iϵI} is a countable ws-O-C { X - Fi:iϵI} has finite sub cover say { X - Fi:i = 1,…,n}. Where, X = ∪i=1 n (X - Fi) ⟺ X = X =∩i=1 n Fi ⟹ X - X = X -∩i=1 n Fi. Hence,  =∩i=1 n Fi.  finite subcollection {Fi.in} of {Fi : iϵI}∋ X = ∪i=1 n (X - Fi). Then,  =∩i=1 n Fi. which contradicts the assumption, ∩i∈I n (Fi)  . Theorem 4.6. In case a map ⴇ : X → Y is ws-irresolute besides B subset of X is ws–Compact related to X, thereafter, the illustration ⴇ ( B ) is ws-Compact related towards Y. Proof: Suppose that {Ģi : i∆0} be multiples of ws-O  Y ∋ ⴇ (B)∪ {Ģi : i∆ }holds. By hypothesis  a finite   B∪ {ⴇ -1(Ģi) : i∆0}. ∴, Here, we have ⴇ (B)  (Ģi : i∆0} , showing that ⴇ (B) are Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 263 https://internationalpubls.com considered as ws-Compact related towards Y. Theorem 4.7. The X × Y of two non-empty space is ws-Compact. Proof. Considering X →Y represent the non-empty spaces' product space X and Y also considering X × Y is a ws - Compact. Subsequently, the projection ∏ ∶ X × Y are considered as ws -irresolute map. ∴, (X × Y) = X is ws - Compact. Likewise, we illustrate aimed at the space Y. 5. ws – Connectedness (Cws) in TS: Definition 5.1. A TS X is named as ws-Cws if X cannot be indicated as a disjoint union of two non - empty ws- O-S. Example 5.2. Ӽ = {L, G, H, K}  = {Ӽ, , {L}, { K }, {L, K}}, ws – CS are { Ӽ, , {G }, { H }, {K }, {G , H}, {G, K }, {H, K }}. ws – O-S are {Ӽ, , {L, H, K }, { L, G, K }, { L, G, H}, {L, K}, { L, H }, { L, G}}. Let U = {L, G} and V = {L, K}. Here X cannot be expressed as the union of two non-empty disjoint ws – O-S. ∴, X is ws – Cws. Theorem 5.3. Every ws-Cws space is Cws . Proof. Considering (Ҳ, ) is known as ws-Cws space. Supposing that (Ҳ, ) is not that Cws. Later, Ҳ = Ģ∪Q, where Ģ and Q be there disjoint nonempty O-S in (Ҳ, ). As it is already known, arbitrary ∪ ws-O-S is ws-O, Ģ and Q are ws - O, Ҳ = Ģ∪Q, where Ģ and Q are disjoint nonempty and ws- O-S in (Ҳ, ). This opposes the point that (Ҳ, ) is ws-Cws and so (Ҳ, ) is Cws. Definition 5.4. A function ⴇ : X →Y is named as ws-irresolute if ⴇ -1(V) is ws-C in X  ws-CS V of Y. Theorem: 5.5 If ⴇ X → Y is a ws - irresolute surjection and X is ws-Cws, hence Y is ws-Cws. Proof. Considering Y is not that ws-Cws, Y = Ģ∪Q where Ģ and Q are disjoint non-empty ws-O-S throughout Y. Subsequently, ⴇ is ws-irresolute besides on, X = ⴇ -1(Ģ) ∪ ⴇ -1 (Q) where ⴇ -1(Ģ) and ⴇ -1(Q) are disjoint non-empty ws-O-S in X. The fact X is contradicted by this ws-Cws. ∴, Y is Cws. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 264 https://internationalpubls.com Remark 5.6. Any indiscrete space with two points is ws – Cws, but any two – point set with discrete topology is not ws- Cws. Theorem 5.7. If the ws-O-S Q and ℘ leads to separation of X besides if Y is ws-Cws subspace of X, then Y exists mainly in Q and ℘. Proof. When Q and ℘ are both ws-O in X the sets Q ∩℘ and ℘∩Y are ws-O in Y these two sets are disjoint ∪ in Y. If both were non-empty, they would represent a division y. ∴, There are only one empty one. ∴, Y has to completely lie in Q or in ℘. 6. Conclusion: The definition of two new concepts of connectedness and compactness ws - (compactness) and ws - (connectedness). An X represents ws – compact if every ws – O – C of X have a finite subcover. The ws - Cws and ws – Compactness fulfilled most of the connectedness and compactness properties in TS. The primary goal of this work was to get a deeper understanding of the consequences of Compactness and Connectedness in Beta weakly semi-CS in TS and suggest future lines of research. We have obtained several noteworthy results from our analysis of beta weakly closed sets. The study has strengthened the theoretical foundations of these concepts by developing and verifying several theorems that show these ideas may hold true in various topological domains. There are a number of interesting avenues for further research in the future. The application of connectedness and compactness in beta weakly semi-CS in TS. The study of compactness and connectedness in beta weakly semi-CS in TS has led to significant discoveries in the TS. References [1] D. Andrijevic, On b-open sets, Math. Vesnik, 48 (1996), No. 1-2, 59-64. DOI.10.12691/tjant-6-6-4 [2] K. Rekha, "**b-Compactness and **b-Connectedness in Topological Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 60, no. 5, pp. 286-293, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V60P542. [3] J. H. Park, θ-b-continuous functions, Acta Math. Hungar,110(2006), No. 4, 347-359. https://doi.org/10.1016/j.mcm.2008.05.007 [4] D.Sivaraj and V.E. 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