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π‘Ίπ’ˆ βˆ— -Compact and π‘Ίπ’ˆ βˆ— -Connected Spaces

This paper introduces the concepts of sg βˆ— -compactness and sg βˆ— -connectedness within topology, generalizing traditional notions of compactness and connectedness. It presents definitions, theorems, and properties associated with these new spaces, exploring their implications and interconnections. The study extends the foundational understanding of topological spaces through these innovative constructs.

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International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 17| 𝑺 π’ˆ βˆ— -Compact and 𝑺 π’ˆ βˆ— -Connected Spaces S. Pious Missier1 , J. Arul Jesti2 1 Associate Professor, PG and Research Department of Mathematics, V.O. Chidambaram College,Thoothukudi, India 2 Research Scholar, PG and Research Department of Mathematics,V.O. Chidambaram College, Thoothukudi, India I. Introduction The notions of compactness and connectedness are useful and fundamental notions of not only general topology but also of other advanced branches of mathematics. Many researchers have investigated the basic properties of compactness and connectedness. The productivity and fruitfulness of these notions of compactness and connectedness motivated mathematicians to generalize these notions. In the course of these attempts many stronger and weaker forms of compactness and connectedness have been introduced and investigated. Recently, S.Pious Missier and J.Arul Jesti have introduced the concept of 𝑆𝑔 βˆ— -open sets[3], and introduced some more functions in 𝑆𝑔 βˆ— -open sets. The aim of this paper is to introduce the concept of 𝑆𝑔 βˆ— -compactness and 𝑆𝑔 βˆ— - connectedness and to investigate some of its characterizations. II. Preliminaries Throughout this paper (X, Ο„), (Y, Οƒ ) and (Z,  ) (or X,Y and Z) represent non-empty topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a space (X, Ο„), 𝑆𝑔 βˆ— C𝑙(A) and 𝑆𝑔 βˆ— 𝐼𝑛𝑑(A) denote the 𝑆𝑔 βˆ— -closure and the 𝑆𝑔 βˆ— -interior of A respectively. Definition 2.1: A subset A of a topological space (X,𝜏) is called a 𝑺 π’ˆ βˆ— -open set [3] if there is an open set U in X such that UβŠ†AβŠ† π‘ πΆπ‘™βˆ— (U). The collection of all 𝑆𝑔 βˆ— -open sets in (X, Ο„) is denoted by 𝑆𝑔 βˆ— O(X,𝜏). Definition 2.2: A subset A of a topological space (X,𝜏) is called a 𝑺 π’ˆ βˆ— -closed set[3] if XA is 𝑆𝑔 βˆ— -open. The collection of all 𝑆𝑔 βˆ— -closed sets in (X, Ο„) is denoted by 𝑆𝑔 βˆ— 𝐢(X,𝜏). Theorem 2.3 [3]:(i)Every open set is 𝑆𝑔 βˆ— -open (ii)Every closed set is 𝑆𝑔 βˆ— -closed set (iii)Every 𝑆𝑔 βˆ— -open set is semi-open Definition 2.4: A topological space 𝑋, 𝜏 is said to be 𝑺 π’ˆ βˆ— -𝑻 𝟏 𝟐 space [4] if every 𝑆𝑔 βˆ— -open set of X is open in X. Definition 2.5: A topological space 𝑋, 𝜏 is said to be 𝑺 π’ˆ βˆ— -locally indiscrete space [5] if every 𝑆𝑔 βˆ— -open set of X is closed in X. Definition 2.6: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be 𝑺 π’ˆ βˆ— -continuous [4] if the inverse image of every open set in Y is 𝑆𝑔 βˆ— -open in X. Defintion 2.7: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be 𝑺 π’ˆ βˆ— -irresolute[4] if the inverse image of every 𝑆𝑔 βˆ— -open set in Y is 𝑆𝑔 βˆ— -open in X. Definition 2.8: A function 𝑓: 𝑋 β†’ π‘Œ is said to be contra- 𝑺 π’ˆ βˆ— -continuous [5] if the inverse image of every open set in Y is 𝑆𝑔 βˆ— -closed in X. Definition 2.9: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be strongly 𝑺 π’ˆ βˆ— -continuous [4]if the inverse image of every 𝑆𝑔 βˆ— - open set in Y is open in X. Definition 2.10: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be perfectly 𝑺 π’ˆ βˆ— -continuous [4]if the inverse image of every 𝑆𝑔 βˆ— - open set in Y is open and closed in X. Definition 2.11: A function 𝑓: (𝑋, 𝜏) β†’ (π‘Œ, 𝜎) is said to be slightly 𝑺 π’ˆ βˆ— -continuous[7] if the inverse image of every clopen set in π‘Œ is 𝑆𝑔 βˆ— -open in 𝑋. ABSTRACT: The determination of this paper is to introduce two new spaces , namely 𝑆𝑔 βˆ— -compact and 𝑆𝑔 βˆ— - connected spaces. Additionally some properties of these spaces are investigated. Mathematics Subject Classification: 54A05 Keywords and phrases: 𝑆𝑔 βˆ— -open set, 𝑆𝑔 βˆ— -closed set, 𝑆𝑔 βˆ— -compact space and 𝑆𝑔 βˆ— -connected space
𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -Connected Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 18| Definition 2.12: A mapping 𝑓: 𝑋, 𝜏 β†’ π‘Œ, 𝜎 is said to be totally 𝑺 π’ˆ βˆ— -continuous[7] if the inverse image of every open set in π‘Œ, 𝜎 is 𝑆𝑔 βˆ— -clopen in 𝑋, 𝜏 . Definition 2.13: A topological space (𝑋, 𝜏) is said to be compact[8](resp. semi-compact[2]) if every open(resp. semi-open) cover of (𝑋, 𝜏) has a finite subcover. Definition 2.14: A topological space 𝑋, 𝜏 is said to be connected[8](resp. semi-connected[1]) if X cannot be expressed as the union of two non-empty open (resp. semi-open) sets in X. III. 𝑺 π’ˆ βˆ— -Compactness Definition 3.1: A collection {𝐴𝑖: 𝑖 ∈ Ξ›} of 𝑆𝑔 βˆ— -open sets in a topological space (𝑋, 𝜏) is called a 𝑺 π’ˆ βˆ— -open cover of a subset 𝐴 in (𝑋, 𝜏) if 𝐴 βŠ‚ 𝐴𝑖.π‘–βˆˆΞ› Definition 3.2:A topological space (𝑋, 𝜏)is called 𝑺 π’ˆ βˆ— -compact if every 𝑆𝑔 βˆ— -open cover of (𝑋, 𝜏) has a finite subcover. Definition 3.3:A subset 𝐴 of a topological space (𝑋, 𝜏) is called 𝑺 π’ˆ βˆ— -compact relative to 𝑿 if for every collection π‘ˆπ‘–: 𝑖 ∈ Ξ› of a 𝑆𝑔 βˆ— -open subsets of 𝑋 such that 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› , there exists a finite subset Ξ›0 of Ξ› such that 𝐴 βŠ‚βˆͺ {π‘ˆπ‘–: 𝑖 ∈ 𝛬0}. Definition 3.4: A subset B of a topological space X is said to be 𝑺 π’ˆ βˆ— -compact if B is 𝑆𝑔 βˆ— -compact as a subspace of X. Theorem 3.5: (i) Every 𝑆𝑔 βˆ— -compact space is compact. (ii) Every semi-compact space is 𝑆𝑔 βˆ— -compact. Proof: (i) and (ii) follows from definition 2.13 and from definition 3.2. Theorem 3.6: Every 𝑆𝑔 βˆ— -closed subset of a 𝑆𝑔 βˆ— - compact space 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact relative to 𝑋, 𝜏 . Proof: Let A be a 𝑆𝑔 βˆ— -closed subset of a 𝑆𝑔 βˆ— - compact space 𝑋, 𝜏 . Then Ac is 𝑆𝑔 βˆ— -open in 𝑋, 𝜏 . Let π‘ˆπ‘–: 𝑖 ∈ Ξ› be a cover of A by 𝑆𝑔 βˆ— -open subsets of X such that 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› . So 𝐴𝑐 βˆͺ π‘ˆπ‘–: 𝑖 ∈ 𝛬 = 𝑋. Since 𝑋, 𝜏 is 𝑆𝑔 βˆ— - there exists a finite subset Ξ›0 of Ξ› such that 𝐴 βŠ‚ 𝐴𝑐 βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› =X. Then 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› and hence A is 𝑆𝑔 βˆ— -compact relative to X. Theorem 3.7: Let 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) be a surjective 𝑆𝑔 βˆ— -continuous map. If 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact, then (π‘Œ, 𝜎) is compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be an open cover of Y. Since 𝑓 is 𝑆𝑔 βˆ— -continuous, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a 𝑆𝑔 βˆ— -open cover of X. Also, since X is 𝑆𝑔 βˆ— -compact, it has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 }. The surjectiveness of 𝑓 implies {𝐴1, 𝐴2, … , 𝐴 𝑛 } is a finite subcover of Y and hence Y is compact. Theorem 3.8: Let 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) be a 𝑆𝑔 βˆ— -irresolute surjective map. If 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact, then (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be a 𝑆𝑔 βˆ— -open cover of Y. Since 𝑓 is 𝑆𝑔 βˆ— -irresolute, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a 𝑆𝑔 βˆ— -open cover of X. Also, since X is 𝑆𝑔 βˆ— -compact, it has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 }. Now 𝑓 is onto implies {𝐴1, 𝐴2,… , 𝐴 𝑛 } is a finite subcover of Y and hence Y is 𝑆𝑔 βˆ— -compact. Theorem 3.9: If a map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -irresolute and a subset 𝐡 of 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact relative to X, then the image 𝑓(𝐡) is 𝑆𝑔 βˆ— -compact relative to π‘Œ. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be any collection of 𝑆𝑔 βˆ— -open subsets of π‘Œ such that 𝑓(𝐡) βŠ‚βˆͺ 𝐴𝑖: 𝑖 ∈ Ξ› . Then 𝐡 βŠ‚βˆͺ π‘“βˆ’1 (𝐴𝑖): 𝑖 ∈ Ξ› holds. Since by hypothesis 𝐡 is 𝑆𝑔 βˆ— -compact relative to 𝑋, there exists a finite subset Ξ›0 of Ξ› such that 𝐡 βŠ‚βˆͺ π‘“βˆ’1 (𝐴𝑖): 𝑖 ∈ Ξ›0 . Therefore we have 𝑓(𝐡) βŠ‚βˆͺ 𝐴𝑖: 𝑖 ∈ Ξ›0 which shows that 𝑓(𝐡) is 𝑆𝑔 βˆ— - compact relative to π‘Œ. Theorem 3.10: If a surjective map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is strongly 𝑆𝑔 βˆ— -continuous and 𝑋, 𝜏 is a compact space, then (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be a 𝑆𝑔 βˆ— -open cover of Y. Since 𝑓 is strongly 𝑆𝑔 βˆ— -continuous, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a open cover of X. Thus the open cover has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 } as X is compact. The surjectiveness of 𝑓 implies {𝐴1, 𝐴2,… , 𝐴 𝑛 } is a finite subcover of Y and hence Y is 𝑆𝑔 βˆ— -compact. Corollary 3.11: If a surjective map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is perfectly 𝑆 𝑔 βˆ— -continuous and 𝑋, 𝜏 is a compact space, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -compact. Proof: Since every perfectly 𝑆 𝑔 βˆ— -continuous function is strongly 𝑆 𝑔 βˆ— -continuous, the result follows from theorem 3.10.
𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -Connected Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 19| IV. 𝑺 π’ˆ βˆ— -Connectedness Definition 4.1: A topological space 𝑋, 𝜏 is called a 𝑆 𝑔 βˆ— -connected space if X cannot be written as a disjoint union of two nonempty 𝑆 𝑔 βˆ— -open sets. Example 4.2: Let 𝑋 = π‘Ž , 𝑏 , 𝑐 with 𝜏 = βˆ…, 𝑋, π‘Ž . Then 𝑆 𝑔 βˆ— 𝑂 𝑋, 𝜏 = βˆ…, 𝑋, π‘Ž , π‘Ž , 𝑏 , { π‘Ž , 𝑐 } and it is 𝑆 𝑔 βˆ— -connected. Theorem 4.3: Every 𝑆 𝑔 βˆ— -connected space is connected. Proof: Let 𝑋, 𝜏 be a 𝑆 𝑔 βˆ— -connected space. Suppose that 𝑋, 𝜏 is not connected, then 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty open sets in 𝑋, 𝜏 . Since every open set is a 𝑆 𝑔 βˆ— -open set, 𝑋, 𝜏 is not a 𝑆 𝑔 βˆ— - connected space and so 𝑋, 𝜏 is connected. Remark 4.4: The converse of theorem 4.3 is not true as can be seen from the following example. Example 4.5: Let 𝑋 = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } and 𝜏 = { 𝑋, πœ™, { π‘Ž }, { 𝑏 , 𝑐 }, { π‘Ž , 𝑏 , 𝑐 }}. Clearly 𝑋, 𝜏 is connected. The 𝑆 𝑔 βˆ— -open sets of 𝑋 are { 𝑋, πœ™, { π‘Ž }, { 𝑏 , 𝑐 }, { π‘Ž , 𝑏 , 𝑐 }, { 𝑏 , 𝑐 , 𝑑 }}. Here 𝑋, 𝜏 is not 𝑆 𝑔 βˆ— -connected because 𝑋 = π‘Ž βˆͺ { 𝑏 , 𝑐 , 𝑑 } where π‘Ž and { 𝑏 , 𝑐 , 𝑑 } are non-empty 𝑆 𝑔 βˆ— -open sets. Theorem 4.6: A contra 𝑆 𝑔 βˆ— -continuous image of a 𝑆 𝑔 βˆ— -connected space is connected. Proof: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a contra 𝑆 𝑔 βˆ— -continuous map of a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto a topological space ( π‘Œ , 𝜎 ). Suppose ( π‘Œ , 𝜎 ) is not connected. Let 𝐴 and 𝐡 form a disconnection of π‘Œ . Then 𝐴 and 𝐡 are clopen and π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 ∩ 𝐡 = βˆ…. Since 𝑓 is contra 𝑆 𝑔 βˆ— -continuous, 𝑋 = 𝑓 βˆ’1 π‘Œ = 𝑓 βˆ’1 𝐴 βˆͺ 𝐡 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡). Thus 𝑋 is the union of disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Also 𝑓 βˆ’1 𝐴 ∩ 𝑓 βˆ’1 𝐡 = βˆ…. Hence 𝑋 is not 𝑆 𝑔 βˆ— -connected which is a contradiction. Therefore π‘Œ is connected. Theorem 4.7: For a subset A of a topological space 𝑋, 𝜏 , the following are equivalent. (i) 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. (ii) The only subsets of 𝑋, 𝜏 which are both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed are the empty set βˆ… and X. (iii) Each 𝑆 𝑔 βˆ— -continuous map of 𝑋, 𝜏 into a discrete space ( π‘Œ , 𝜎 ) with atleast two points is a constant map. Proof: (i)⟹(ii): Suppose that 𝑆 βŠ‚ 𝑋 is a proper subset, which is both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed. Then 𝑆 𝑐 is also 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed. Therefore 𝑋 = 𝑆 βˆͺ 𝑆 𝑐 is a disjoint union of two nonempty 𝑆 𝑔 βˆ— -open sets which contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Hence 𝑆 = βˆ… or 𝑆 = 𝑋. (ii) ⟹(i): Suppose that 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since 𝐴 = 𝐡 𝑐 , A is 𝑆 𝑔 βˆ— -closed. But by assumption 𝐴 = βˆ…, which is a contradiction. Hence (i) holds. (ii) ⟹ (iii): Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a 𝑆 𝑔 βˆ— -continuous map where ( π‘Œ , 𝜎 ) is a discrete space with atleast two points. Then 𝑓 βˆ’1 ( 𝑦 ) is 𝑆 𝑔 βˆ— -closed and 𝑆 𝑔 βˆ— -open for each 𝑦 ∈ π‘Œ and 𝑋 =βˆͺ { 𝑓 βˆ’1 𝑦 : 𝑦 ∈ π‘Œ }. By assumption, 𝑓 βˆ’1 𝑦 = βˆ… or 𝑓 βˆ’1 𝑦 = 𝑋. If 𝑓 βˆ’1 𝑦 = βˆ… for all 𝑦 ∈ π‘Œ , then 𝑓 will not be a map. Hence, there exists only one point say 𝑦 1 ∈ π‘Œ such that 𝑓 βˆ’1 𝑦 β‰  βˆ… and 𝑓 βˆ’1 𝑦 1 = 𝑋 which shows that f is a constant map. (iii) ⟹(ii):Let π‘ˆ be both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed in 𝑋, 𝜏 . Suppose that π‘ˆ β‰  βˆ…. Define 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) by 𝑓 π‘ˆ = { 𝑦 1} and 𝑓 ( π‘ˆ 𝑐 ) = { 𝑦 2} for some distinct points 𝑦 1 and 𝑦 2 in ( π‘Œ , 𝜎 ), then 𝑓 is 𝑆 𝑔 βˆ— -continuous. By assumption, 𝑓 is a constant map. Therefore 𝑦 1 = 𝑦 2 and so π‘ˆ = 𝑋. Theorem 4.8: Let 𝑋, 𝜏 be 𝑆 𝑔 βˆ— -connected. Then each contra 𝑆 𝑔 βˆ— -continuous map of 𝑋, 𝜏 into a discrete space ( π‘Œ , 𝜎 ) with atleast two points is a constant map. Proof: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a contra 𝑆 𝑔 βˆ— -continuous map where ( π‘Œ , 𝜎 ) is a discrete space with atleast two points. Then 𝑋 is covered by 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed covering { 𝑓 βˆ’1 𝑦 : 𝑦 ∈ π‘Œ }. Since 𝑋, 𝜏 is 𝑆 𝑔 βˆ— - connected, the only subsets of 𝑋, 𝜏 which are both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed are the empty set βˆ… and X. Therefore 𝑓 βˆ’1 𝑦 = βˆ… or 𝑓 βˆ’1 𝑦 = 𝑋. If 𝑓 βˆ’1 𝑦 = βˆ… for all 𝑦 ∈ π‘Œ , then 𝑓 fails to be a map. Then, there exists only one point say 𝑦 ∈ π‘Œ such that 𝑓 βˆ’1 𝑦 β‰  βˆ… and 𝑓 βˆ’1 𝑦 = 𝑋 which shows that f is a constant map. Theorem 4.9: If 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is a 𝑆 𝑔 βˆ— -continuous surjection and 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected, then ( π‘Œ , 𝜎 ) is connected. Proof: Suppose ( π‘Œ , 𝜎 ) is not connected. Then π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 and 𝐡 are disjoint nonempty open subsets of ( π‘Œ , 𝜎 ). Since 𝑓 is 𝑆 𝑔 βˆ— -continuous and onto, 𝑋 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡) where 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . This contradicts the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected and hence ( π‘Œ , 𝜎 ) is connected. Theorem 4.10: If a surjective map 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -irresolute and 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Proof: If possible assume that π‘Œ is not 𝑆 𝑔 βˆ— -connected. Then π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 and 𝐡 are nonempty disjoint 𝑆 𝑔 βˆ— -open sets of ( π‘Œ , 𝜎 ). Since 𝑓 is 𝑆 𝑔 βˆ— -irresolute, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since 𝑓 is onto, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are nonempty. Now 𝑋 = 𝑓 βˆ’1 π‘Œ = 𝑓 βˆ’1 𝐴 βˆͺ 𝐡 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡). Thus 𝑋 is
𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -Connected Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 20| the union of disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . This contradicts the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected and hence ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Theorem 4.11: If a surjective map 𝑓 : 𝑋, 𝜏 β†’ π‘Œ , 𝜎 is strongly 𝑆 𝑔 βˆ— -continuous and 𝑋, 𝜏 is a connected space, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Proof: Similar to the proof of the above theorem. Theorem 4.12: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a perfectly 𝑆 𝑔 βˆ— -continuous map, 𝑋, 𝜏 a connected space, then ( π‘Œ , 𝜎 ) has an indiscrete topology. Proof: Suppose that there exists a proper open set U of ( π‘Œ , 𝜎 ), then U is 𝑆 𝑔 βˆ— -open in ( π‘Œ , 𝜎 ). Since 𝑓 is perfectly 𝑆 𝑔 βˆ— -continuous, 𝑓 βˆ’1 ( π‘ˆ) is a proper open and closed subset of 𝑋, 𝜏 . This implies 𝑋, 𝜏 is not connected which is a contradiction. Therefore ( π‘Œ , 𝜎 ) has an indiscrete topology. Theorem 4.13: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a totally 𝑆 𝑔 βˆ— -continuous map, from a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto any space ( π‘Œ , 𝜎 ), then ( π‘Œ , 𝜎 ) is an indiscrete space. Proof: Suppose ( π‘Œ , 𝜎 ) is not indiscrete. Let A be a proper nonempty open subset of ( π‘Œ , 𝜎 ). Then 𝑓 βˆ’1 ( 𝐴) is a proper nonempty 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed subset of 𝑋, 𝜏 , which is a contradiction to the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Then ( π‘Œ , 𝜎 ) must be indiscrete. Theorem 4.14: If 𝑓 is a contra 𝑆 𝑔 βˆ— -continuous map from a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto any space ( π‘Œ , 𝜎 ), then ( π‘Œ , 𝜎 ) is not a discrete space. Proof: Suppose ( π‘Œ , 𝜎 ) is discrete. Let A be any proper nonempty open and closed subset of ( π‘Œ , 𝜎 ). Then 𝑓 βˆ’1 ( 𝐴) is a proper nonempty 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed subset of 𝑋, 𝜏 , which is a contradiction to the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Hence (π‘Œ, 𝜎 ) is not a discrete space. Theorem 4.15: Suppose that 𝑋 is a 𝑆 𝑔 βˆ— - 𝑇 1 2 space then 𝑋 is connected if and only if it is 𝑆 𝑔 βˆ— -connected. Proof: Suppose that 𝑋 is connected. Then 𝑋 cannot be written as a union of two non-empty disjoint proper subsets of X. Suppose 𝑋 is not 𝑆 𝑔 βˆ— -connected. Let 𝐴 and 𝐡 be any two 𝑆 𝑔 βˆ— -open sets subsets of 𝑋 such that 𝑋 = 𝐴 βˆͺ 𝐡, where 𝐴 ∩ 𝐡 = βˆ…. Since 𝑋 is a 𝑆 𝑔 βˆ— - 𝑇 1 2 space, every 𝑆 𝑔 βˆ— -open sets are open. Hence 𝐴 and 𝐡 are open sets which contradicts the fact that 𝑋 is not connected. Then 𝑋 is 𝑆 𝑔 βˆ— -connected. The converse part follows from the theorem that every 𝑆 𝑔 βˆ— -connected space is connected. Theorem 4.16: If 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is slightly 𝑆 𝑔 βˆ— -continuous surjective function and X is 𝑆 𝑔 βˆ— -connected then Y is connected. Proof: Suppose Y is not connected. Then there exists non-empty disjoint open set A and B such that Y= 𝐴 βˆͺ 𝐡. Therefore A and B are clopen sets in Y. Since 𝑓 is slightly 𝑆 𝑔 βˆ— -continuous and surjective, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡)are non-empty disjoint 𝑆 𝑔 βˆ— -opensets in X. Also 𝑓 βˆ’1 π‘Œ = 𝑋 = 𝑓 βˆ’1 𝐡 . This shows that X is not 𝑆 𝑔 βˆ— - connected, a contradiction. Hence Y is connected. Theorem 4.17: If 𝑓 is slightly 𝑆 𝑔 βˆ— -continuous function from a connected space ( 𝑋, 𝜏 ) onto a space ( π‘Œ , 𝜎 ) then Y is not a discrete space. Proof: Suppose that Y is a discrete space. Let A be a proper nonempty open subset of Y. Then 𝑓 βˆ’1 ( 𝐴) is nonempty 𝑆 𝑔 βˆ— -clopen subset of X, which is a contradiction to the fact that X is 𝑆 𝑔 βˆ— -connected. Hence Y is not a discrete space. Theorem 4.18: A space X is 𝑆 𝑔 βˆ— -connected if every slightly 𝑆 𝑔 βˆ— -continuous from X into any 𝑇 π‘œ space Y is constant. Proof: Let every slightly 𝑆 𝑔 βˆ— -continuous function from a space X into Y be constant. If X is not 𝑆 𝑔 βˆ— -connected then there exists a proper nonempty 𝑆 𝑔 βˆ— -clopen subset A of X . Let ( π‘Œ , 𝜎 ) be such that Y= π‘Ž , 𝑏 𝜎 = {βˆ…, 𝑦 , π‘Ž , 𝑏 } be a topology. Let 𝑓 : 𝑋 β†’ π‘Œ be any function such that 𝑓 𝐴 = { π‘Ž } and 𝑓 𝑋 βˆ’ 𝐴 = 𝑏 . Then f is a non-constant and slightly 𝑆 𝑔 βˆ— -continuous function such that Y is 𝑇 π‘œ which is a contradiction. Hence Y is 𝑆 𝑔 βˆ— -connected. Theorem 4.19: A space 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected if and only if every totally 𝑆 𝑔 βˆ— -continuous function from a space 𝑋, 𝜏 into any 𝑇 0 space ( π‘Œ , 𝜎 ) is a constant map. Proof: Suppose 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is a totally 𝑆 𝑔 βˆ— -continuous function where ( π‘Œ , 𝜎 ) is a 𝑇 0-space. Suppose that f is not a constant map, then we can select two points π‘₯ and 𝑦 such that 𝑓 π‘₯ β‰  𝑓 𝑦 . Since ( π‘Œ , 𝜎 ) is a 𝑇 0-spaceand 𝑓 ( π‘₯ ) and 𝑓 ( 𝑦 ) are distinct points of Y,there exists an open set G in ( π‘Œ , 𝜎 ) containing 𝑓 ( π‘₯ ) but not 𝑓 ( 𝑦 ). Since 𝑓 is a totally 𝑆 𝑔 βˆ— -continuous function, 𝑓 βˆ’1 ( 𝐺) is a 𝑆 𝑔 βˆ— -clopen subset of π‘₯ , 𝜏 . Clearly π‘₯ ∈ 𝑓 βˆ’1 ( 𝐺) and 𝑦 βˆ‰ 𝑓 βˆ’1 𝐺 . Now 𝑋 = 𝑓 βˆ’1 𝐺 βˆͺ 𝑓 βˆ’1 𝐺 c which is the union of non-empty 𝑆 𝑔 βˆ— -open subsets of X. Thus X is not 𝑆 𝑔 βˆ— -connected space,which contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Hence f is a constant map. Conversely, suppose ( 𝑋, 𝜏 ) is not a 𝑆 𝑔 βˆ— -connected space there exists a proper non-empty 𝑆 𝑔 βˆ— -clopen subset 𝐴 of 𝑋. Let π‘Œ = { π‘Ž , 𝑏 } and 𝜏 = π‘Œ , βˆ…, π‘Ž , 𝑏 be a topology for π‘Œ . Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a function
𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -Connected Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 21| such that 𝑓 𝐴 = { π‘Ž } and 𝑓 π‘Œ 𝐴 = 𝑏 . Then 𝑓 is non-constant and totally 𝑆 𝑔 βˆ— -continuous such that π‘Œ is 𝑇 0, which is a contradiction. Hence X must be 𝑆 𝑔 βˆ— -connected. Theorem 4.20: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a totally 𝑆 𝑔 βˆ— -continuous function & π‘Œ is a 𝑇 1-space. If A is a non- empty 𝑆 𝑔 βˆ— -connected subset of X. Then 𝑓 ( 𝐴)is singleton. Proof: Suppose that 𝑓 ( 𝐴)is not a singleton . Let 𝑓 ( π‘₯ 1 ) = 𝑦 1 ∈ 𝐴 and 𝑓 ( π‘₯ 2 ) = 𝑦 2 ∈ 𝐴. Since 𝑦 1, 𝑦 2 ∈ π‘Œ and Y is a 𝑇 1-space, there exists an open set G in ( π‘Œ , 𝜎 ) containing y,but not 𝑦 2. Since 𝑓 is totally 𝑆 𝑔 βˆ— - continuous, 𝑓 βˆ’1 ( 𝐺) is 𝑆 𝑔 βˆ— -continuous, 𝑓 βˆ’1 ( 𝐺) is 𝑆 𝑔 βˆ— -clopen set containing π‘₯ 1 but not π‘₯ 2. Now 𝑋 = 𝑓 βˆ’1 𝐺 βˆͺ 𝑓 βˆ’1 𝐺 c . Thus we have expressed X as a union of two non-empty 𝑆 𝑔 βˆ— -open sets. This contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Therefore 𝑓 ( 𝐴) is singleton. Theorem 4.21: Every semi-connected space is 𝑆 𝑔 βˆ— -connected. Proof: Let 𝑋, 𝜏 be a semi-connected space. Suppose that 𝑋, 𝜏 is not 𝑆 𝑔 βˆ— -connected, then 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since every 𝑆 𝑔 βˆ— -open set is semi-open, 𝑋, 𝜏 is not a semi-connected space which is a contradiction and hence 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Remark 4.22: The converse of theorem 4.21 is not true as can be seen from the following example. Example 4.23: Let 𝑋 = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } and 𝜏 = { 𝑋, πœ™, π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑏 , 𝑏 , 𝑐 , π‘Ž , 𝑐 , { π‘Ž , 𝑏 , 𝑐 }}. Then 𝑆𝑂 𝑋, 𝜏 = { 𝑋, πœ™, π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑏 , 𝑏 , 𝑐 , π‘Ž , 𝑐 , π‘Ž , 𝑑 , 𝑏 , 𝑑 , 𝑐 , 𝑑 , π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑐 , 𝑑 , π‘Ž , 𝑏 , 𝑑 , 𝑏 , 𝑐 , 𝑑 } and 𝑆 𝑔 βˆ— 𝑂 𝑋, 𝜏 = 𝜏 Here 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected but not semi-connected because 𝑋 = π‘Ž βˆͺ { 𝑏 , 𝑐 , 𝑑 } where π‘Ž and { 𝑏 , 𝑐 , 𝑑 } are non-empty disjoint semi-open sets. REFERENCES [1] Das.P., I.J.M.M. 12,1974,31-34. [2] Dorsett.C., Semi compactness, semi separation axioms and product spaces,Bull. Of Malaysian Mathematical sciences Society, 4(1), 1981, 21-28. [3] Pious Missier.S and Arul Jesti.J. , A new notion of open sets in topological spaces, Int.J.Math.Arc., 3(11), 2012, 3990-3996. [4] Pious Missier.S and Arul Jesti.J., Some new functions in topological spaces, Outreach(A multi-disciplinary research Journal), 7, 2014,161-168. [5] Pious Missier.S and Arul Jesti.J., Properties of 𝑆 𝑔 βˆ— -functions in topological spaces, Math. Sci. Int. Research Journal(IMRF), 3(2), 2014, 911-915. [6] Pious Missier.S and Arul Jesti.J., 𝑆 𝑔 βˆ— -homeomorphism in topological spaces, Outreach(A multi-disciplinary research Journal) (Communicated). [7] Pious Missier.S and Arul Jesti.J., Slightly 𝑆 𝑔 βˆ— -continuous functions and Totally 𝑆 𝑔 βˆ— -continuous functions in Topological spaces,IOSR-JM, (Accepted). [8] Willard S., General topology, Addison Wesley, 1970.

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π‘Ίπ’ˆ βˆ— -Compact and π‘Ίπ’ˆ βˆ— -Connected Spaces

  • 1.
    International OPEN ACCESS Journal OfModern Engineering Research (IJMER) | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 17| 𝑺 π’ˆ βˆ— -Compact and 𝑺 π’ˆ βˆ— -Connected Spaces S. Pious Missier1 , J. Arul Jesti2 1 Associate Professor, PG and Research Department of Mathematics, V.O. Chidambaram College,Thoothukudi, India 2 Research Scholar, PG and Research Department of Mathematics,V.O. Chidambaram College, Thoothukudi, India I. Introduction The notions of compactness and connectedness are useful and fundamental notions of not only general topology but also of other advanced branches of mathematics. Many researchers have investigated the basic properties of compactness and connectedness. The productivity and fruitfulness of these notions of compactness and connectedness motivated mathematicians to generalize these notions. In the course of these attempts many stronger and weaker forms of compactness and connectedness have been introduced and investigated. Recently, S.Pious Missier and J.Arul Jesti have introduced the concept of 𝑆𝑔 βˆ— -open sets[3], and introduced some more functions in 𝑆𝑔 βˆ— -open sets. The aim of this paper is to introduce the concept of 𝑆𝑔 βˆ— -compactness and 𝑆𝑔 βˆ— - connectedness and to investigate some of its characterizations. II. Preliminaries Throughout this paper (X, Ο„), (Y, Οƒ ) and (Z,  ) (or X,Y and Z) represent non-empty topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a space (X, Ο„), 𝑆𝑔 βˆ— C𝑙(A) and 𝑆𝑔 βˆ— 𝐼𝑛𝑑(A) denote the 𝑆𝑔 βˆ— -closure and the 𝑆𝑔 βˆ— -interior of A respectively. Definition 2.1: A subset A of a topological space (X,𝜏) is called a 𝑺 π’ˆ βˆ— -open set [3] if there is an open set U in X such that UβŠ†AβŠ† π‘ πΆπ‘™βˆ— (U). The collection of all 𝑆𝑔 βˆ— -open sets in (X, Ο„) is denoted by 𝑆𝑔 βˆ— O(X,𝜏). Definition 2.2: A subset A of a topological space (X,𝜏) is called a 𝑺 π’ˆ βˆ— -closed set[3] if XA is 𝑆𝑔 βˆ— -open. The collection of all 𝑆𝑔 βˆ— -closed sets in (X, Ο„) is denoted by 𝑆𝑔 βˆ— 𝐢(X,𝜏). Theorem 2.3 [3]:(i)Every open set is 𝑆𝑔 βˆ— -open (ii)Every closed set is 𝑆𝑔 βˆ— -closed set (iii)Every 𝑆𝑔 βˆ— -open set is semi-open Definition 2.4: A topological space 𝑋, 𝜏 is said to be 𝑺 π’ˆ βˆ— -𝑻 𝟏 𝟐 space [4] if every 𝑆𝑔 βˆ— -open set of X is open in X. Definition 2.5: A topological space 𝑋, 𝜏 is said to be 𝑺 π’ˆ βˆ— -locally indiscrete space [5] if every 𝑆𝑔 βˆ— -open set of X is closed in X. Definition 2.6: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be 𝑺 π’ˆ βˆ— -continuous [4] if the inverse image of every open set in Y is 𝑆𝑔 βˆ— -open in X. Defintion 2.7: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be 𝑺 π’ˆ βˆ— -irresolute[4] if the inverse image of every 𝑆𝑔 βˆ— -open set in Y is 𝑆𝑔 βˆ— -open in X. Definition 2.8: A function 𝑓: 𝑋 β†’ π‘Œ is said to be contra- 𝑺 π’ˆ βˆ— -continuous [5] if the inverse image of every open set in Y is 𝑆𝑔 βˆ— -closed in X. Definition 2.9: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be strongly 𝑺 π’ˆ βˆ— -continuous [4]if the inverse image of every 𝑆𝑔 βˆ— - open set in Y is open in X. Definition 2.10: A mapping 𝑓: 𝑋 β†’ π‘Œ is said to be perfectly 𝑺 π’ˆ βˆ— -continuous [4]if the inverse image of every 𝑆𝑔 βˆ— - open set in Y is open and closed in X. Definition 2.11: A function 𝑓: (𝑋, 𝜏) β†’ (π‘Œ, 𝜎) is said to be slightly 𝑺 π’ˆ βˆ— -continuous[7] if the inverse image of every clopen set in π‘Œ is 𝑆𝑔 βˆ— -open in 𝑋. ABSTRACT: The determination of this paper is to introduce two new spaces , namely 𝑆𝑔 βˆ— -compact and 𝑆𝑔 βˆ— - connected spaces. Additionally some properties of these spaces are investigated. Mathematics Subject Classification: 54A05 Keywords and phrases: 𝑆𝑔 βˆ— -open set, 𝑆𝑔 βˆ— -closed set, 𝑆𝑔 βˆ— -compact space and 𝑆𝑔 βˆ— -connected space
  • 2.
    𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -ConnectedSpaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 18| Definition 2.12: A mapping 𝑓: 𝑋, 𝜏 β†’ π‘Œ, 𝜎 is said to be totally 𝑺 π’ˆ βˆ— -continuous[7] if the inverse image of every open set in π‘Œ, 𝜎 is 𝑆𝑔 βˆ— -clopen in 𝑋, 𝜏 . Definition 2.13: A topological space (𝑋, 𝜏) is said to be compact[8](resp. semi-compact[2]) if every open(resp. semi-open) cover of (𝑋, 𝜏) has a finite subcover. Definition 2.14: A topological space 𝑋, 𝜏 is said to be connected[8](resp. semi-connected[1]) if X cannot be expressed as the union of two non-empty open (resp. semi-open) sets in X. III. 𝑺 π’ˆ βˆ— -Compactness Definition 3.1: A collection {𝐴𝑖: 𝑖 ∈ Ξ›} of 𝑆𝑔 βˆ— -open sets in a topological space (𝑋, 𝜏) is called a 𝑺 π’ˆ βˆ— -open cover of a subset 𝐴 in (𝑋, 𝜏) if 𝐴 βŠ‚ 𝐴𝑖.π‘–βˆˆΞ› Definition 3.2:A topological space (𝑋, 𝜏)is called 𝑺 π’ˆ βˆ— -compact if every 𝑆𝑔 βˆ— -open cover of (𝑋, 𝜏) has a finite subcover. Definition 3.3:A subset 𝐴 of a topological space (𝑋, 𝜏) is called 𝑺 π’ˆ βˆ— -compact relative to 𝑿 if for every collection π‘ˆπ‘–: 𝑖 ∈ Ξ› of a 𝑆𝑔 βˆ— -open subsets of 𝑋 such that 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› , there exists a finite subset Ξ›0 of Ξ› such that 𝐴 βŠ‚βˆͺ {π‘ˆπ‘–: 𝑖 ∈ 𝛬0}. Definition 3.4: A subset B of a topological space X is said to be 𝑺 π’ˆ βˆ— -compact if B is 𝑆𝑔 βˆ— -compact as a subspace of X. Theorem 3.5: (i) Every 𝑆𝑔 βˆ— -compact space is compact. (ii) Every semi-compact space is 𝑆𝑔 βˆ— -compact. Proof: (i) and (ii) follows from definition 2.13 and from definition 3.2. Theorem 3.6: Every 𝑆𝑔 βˆ— -closed subset of a 𝑆𝑔 βˆ— - compact space 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact relative to 𝑋, 𝜏 . Proof: Let A be a 𝑆𝑔 βˆ— -closed subset of a 𝑆𝑔 βˆ— - compact space 𝑋, 𝜏 . Then Ac is 𝑆𝑔 βˆ— -open in 𝑋, 𝜏 . Let π‘ˆπ‘–: 𝑖 ∈ Ξ› be a cover of A by 𝑆𝑔 βˆ— -open subsets of X such that 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› . So 𝐴𝑐 βˆͺ π‘ˆπ‘–: 𝑖 ∈ 𝛬 = 𝑋. Since 𝑋, 𝜏 is 𝑆𝑔 βˆ— - there exists a finite subset Ξ›0 of Ξ› such that 𝐴 βŠ‚ 𝐴𝑐 βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› =X. Then 𝐴 βŠ‚βˆͺ π‘ˆπ‘–: 𝑖 ∈ Ξ› and hence A is 𝑆𝑔 βˆ— -compact relative to X. Theorem 3.7: Let 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) be a surjective 𝑆𝑔 βˆ— -continuous map. If 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact, then (π‘Œ, 𝜎) is compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be an open cover of Y. Since 𝑓 is 𝑆𝑔 βˆ— -continuous, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a 𝑆𝑔 βˆ— -open cover of X. Also, since X is 𝑆𝑔 βˆ— -compact, it has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 }. The surjectiveness of 𝑓 implies {𝐴1, 𝐴2, … , 𝐴 𝑛 } is a finite subcover of Y and hence Y is compact. Theorem 3.8: Let 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) be a 𝑆𝑔 βˆ— -irresolute surjective map. If 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact, then (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be a 𝑆𝑔 βˆ— -open cover of Y. Since 𝑓 is 𝑆𝑔 βˆ— -irresolute, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a 𝑆𝑔 βˆ— -open cover of X. Also, since X is 𝑆𝑔 βˆ— -compact, it has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 }. Now 𝑓 is onto implies {𝐴1, 𝐴2,… , 𝐴 𝑛 } is a finite subcover of Y and hence Y is 𝑆𝑔 βˆ— -compact. Theorem 3.9: If a map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -irresolute and a subset 𝐡 of 𝑋, 𝜏 is 𝑆𝑔 βˆ— -compact relative to X, then the image 𝑓(𝐡) is 𝑆𝑔 βˆ— -compact relative to π‘Œ. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be any collection of 𝑆𝑔 βˆ— -open subsets of π‘Œ such that 𝑓(𝐡) βŠ‚βˆͺ 𝐴𝑖: 𝑖 ∈ Ξ› . Then 𝐡 βŠ‚βˆͺ π‘“βˆ’1 (𝐴𝑖): 𝑖 ∈ Ξ› holds. Since by hypothesis 𝐡 is 𝑆𝑔 βˆ— -compact relative to 𝑋, there exists a finite subset Ξ›0 of Ξ› such that 𝐡 βŠ‚βˆͺ π‘“βˆ’1 (𝐴𝑖): 𝑖 ∈ Ξ›0 . Therefore we have 𝑓(𝐡) βŠ‚βˆͺ 𝐴𝑖: 𝑖 ∈ Ξ›0 which shows that 𝑓(𝐡) is 𝑆𝑔 βˆ— - compact relative to π‘Œ. Theorem 3.10: If a surjective map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is strongly 𝑆𝑔 βˆ— -continuous and 𝑋, 𝜏 is a compact space, then (π‘Œ, 𝜎) is 𝑆𝑔 βˆ— -compact. Proof: Let 𝐴𝑖: 𝑖 ∈ Ξ› be a 𝑆𝑔 βˆ— -open cover of Y. Since 𝑓 is strongly 𝑆𝑔 βˆ— -continuous, {π‘“βˆ’1 𝐴𝑖 : π‘–πœ–Ξ›} is a open cover of X. Thus the open cover has a finite subcover, say {π‘“βˆ’1 𝐴1 , π‘“βˆ’1 𝐴2 , … π‘“βˆ’1 𝐴 𝑛 } as X is compact. The surjectiveness of 𝑓 implies {𝐴1, 𝐴2,… , 𝐴 𝑛 } is a finite subcover of Y and hence Y is 𝑆𝑔 βˆ— -compact. Corollary 3.11: If a surjective map 𝑓: 𝑋, 𝜏 β†’ (π‘Œ, 𝜎) is perfectly 𝑆 𝑔 βˆ— -continuous and 𝑋, 𝜏 is a compact space, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -compact. Proof: Since every perfectly 𝑆 𝑔 βˆ— -continuous function is strongly 𝑆 𝑔 βˆ— -continuous, the result follows from theorem 3.10.
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    𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -ConnectedSpaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 19| IV. 𝑺 π’ˆ βˆ— -Connectedness Definition 4.1: A topological space 𝑋, 𝜏 is called a 𝑆 𝑔 βˆ— -connected space if X cannot be written as a disjoint union of two nonempty 𝑆 𝑔 βˆ— -open sets. Example 4.2: Let 𝑋 = π‘Ž , 𝑏 , 𝑐 with 𝜏 = βˆ…, 𝑋, π‘Ž . Then 𝑆 𝑔 βˆ— 𝑂 𝑋, 𝜏 = βˆ…, 𝑋, π‘Ž , π‘Ž , 𝑏 , { π‘Ž , 𝑐 } and it is 𝑆 𝑔 βˆ— -connected. Theorem 4.3: Every 𝑆 𝑔 βˆ— -connected space is connected. Proof: Let 𝑋, 𝜏 be a 𝑆 𝑔 βˆ— -connected space. Suppose that 𝑋, 𝜏 is not connected, then 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty open sets in 𝑋, 𝜏 . Since every open set is a 𝑆 𝑔 βˆ— -open set, 𝑋, 𝜏 is not a 𝑆 𝑔 βˆ— - connected space and so 𝑋, 𝜏 is connected. Remark 4.4: The converse of theorem 4.3 is not true as can be seen from the following example. Example 4.5: Let 𝑋 = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } and 𝜏 = { 𝑋, πœ™, { π‘Ž }, { 𝑏 , 𝑐 }, { π‘Ž , 𝑏 , 𝑐 }}. Clearly 𝑋, 𝜏 is connected. The 𝑆 𝑔 βˆ— -open sets of 𝑋 are { 𝑋, πœ™, { π‘Ž }, { 𝑏 , 𝑐 }, { π‘Ž , 𝑏 , 𝑐 }, { 𝑏 , 𝑐 , 𝑑 }}. Here 𝑋, 𝜏 is not 𝑆 𝑔 βˆ— -connected because 𝑋 = π‘Ž βˆͺ { 𝑏 , 𝑐 , 𝑑 } where π‘Ž and { 𝑏 , 𝑐 , 𝑑 } are non-empty 𝑆 𝑔 βˆ— -open sets. Theorem 4.6: A contra 𝑆 𝑔 βˆ— -continuous image of a 𝑆 𝑔 βˆ— -connected space is connected. Proof: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a contra 𝑆 𝑔 βˆ— -continuous map of a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto a topological space ( π‘Œ , 𝜎 ). Suppose ( π‘Œ , 𝜎 ) is not connected. Let 𝐴 and 𝐡 form a disconnection of π‘Œ . Then 𝐴 and 𝐡 are clopen and π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 ∩ 𝐡 = βˆ…. Since 𝑓 is contra 𝑆 𝑔 βˆ— -continuous, 𝑋 = 𝑓 βˆ’1 π‘Œ = 𝑓 βˆ’1 𝐴 βˆͺ 𝐡 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡). Thus 𝑋 is the union of disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Also 𝑓 βˆ’1 𝐴 ∩ 𝑓 βˆ’1 𝐡 = βˆ…. Hence 𝑋 is not 𝑆 𝑔 βˆ— -connected which is a contradiction. Therefore π‘Œ is connected. Theorem 4.7: For a subset A of a topological space 𝑋, 𝜏 , the following are equivalent. (i) 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. (ii) The only subsets of 𝑋, 𝜏 which are both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed are the empty set βˆ… and X. (iii) Each 𝑆 𝑔 βˆ— -continuous map of 𝑋, 𝜏 into a discrete space ( π‘Œ , 𝜎 ) with atleast two points is a constant map. Proof: (i)⟹(ii): Suppose that 𝑆 βŠ‚ 𝑋 is a proper subset, which is both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed. Then 𝑆 𝑐 is also 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed. Therefore 𝑋 = 𝑆 βˆͺ 𝑆 𝑐 is a disjoint union of two nonempty 𝑆 𝑔 βˆ— -open sets which contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Hence 𝑆 = βˆ… or 𝑆 = 𝑋. (ii) ⟹(i): Suppose that 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since 𝐴 = 𝐡 𝑐 , A is 𝑆 𝑔 βˆ— -closed. But by assumption 𝐴 = βˆ…, which is a contradiction. Hence (i) holds. (ii) ⟹ (iii): Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a 𝑆 𝑔 βˆ— -continuous map where ( π‘Œ , 𝜎 ) is a discrete space with atleast two points. Then 𝑓 βˆ’1 ( 𝑦 ) is 𝑆 𝑔 βˆ— -closed and 𝑆 𝑔 βˆ— -open for each 𝑦 ∈ π‘Œ and 𝑋 =βˆͺ { 𝑓 βˆ’1 𝑦 : 𝑦 ∈ π‘Œ }. By assumption, 𝑓 βˆ’1 𝑦 = βˆ… or 𝑓 βˆ’1 𝑦 = 𝑋. If 𝑓 βˆ’1 𝑦 = βˆ… for all 𝑦 ∈ π‘Œ , then 𝑓 will not be a map. Hence, there exists only one point say 𝑦 1 ∈ π‘Œ such that 𝑓 βˆ’1 𝑦 β‰  βˆ… and 𝑓 βˆ’1 𝑦 1 = 𝑋 which shows that f is a constant map. (iii) ⟹(ii):Let π‘ˆ be both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed in 𝑋, 𝜏 . Suppose that π‘ˆ β‰  βˆ…. Define 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) by 𝑓 π‘ˆ = { 𝑦 1} and 𝑓 ( π‘ˆ 𝑐 ) = { 𝑦 2} for some distinct points 𝑦 1 and 𝑦 2 in ( π‘Œ , 𝜎 ), then 𝑓 is 𝑆 𝑔 βˆ— -continuous. By assumption, 𝑓 is a constant map. Therefore 𝑦 1 = 𝑦 2 and so π‘ˆ = 𝑋. Theorem 4.8: Let 𝑋, 𝜏 be 𝑆 𝑔 βˆ— -connected. Then each contra 𝑆 𝑔 βˆ— -continuous map of 𝑋, 𝜏 into a discrete space ( π‘Œ , 𝜎 ) with atleast two points is a constant map. Proof: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a contra 𝑆 𝑔 βˆ— -continuous map where ( π‘Œ , 𝜎 ) is a discrete space with atleast two points. Then 𝑋 is covered by 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed covering { 𝑓 βˆ’1 𝑦 : 𝑦 ∈ π‘Œ }. Since 𝑋, 𝜏 is 𝑆 𝑔 βˆ— - connected, the only subsets of 𝑋, 𝜏 which are both 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed are the empty set βˆ… and X. Therefore 𝑓 βˆ’1 𝑦 = βˆ… or 𝑓 βˆ’1 𝑦 = 𝑋. If 𝑓 βˆ’1 𝑦 = βˆ… for all 𝑦 ∈ π‘Œ , then 𝑓 fails to be a map. Then, there exists only one point say 𝑦 ∈ π‘Œ such that 𝑓 βˆ’1 𝑦 β‰  βˆ… and 𝑓 βˆ’1 𝑦 = 𝑋 which shows that f is a constant map. Theorem 4.9: If 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is a 𝑆 𝑔 βˆ— -continuous surjection and 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected, then ( π‘Œ , 𝜎 ) is connected. Proof: Suppose ( π‘Œ , 𝜎 ) is not connected. Then π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 and 𝐡 are disjoint nonempty open subsets of ( π‘Œ , 𝜎 ). Since 𝑓 is 𝑆 𝑔 βˆ— -continuous and onto, 𝑋 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡) where 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . This contradicts the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected and hence ( π‘Œ , 𝜎 ) is connected. Theorem 4.10: If a surjective map 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -irresolute and 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Proof: If possible assume that π‘Œ is not 𝑆 𝑔 βˆ— -connected. Then π‘Œ = 𝐴 βˆͺ 𝐡 where 𝐴 and 𝐡 are nonempty disjoint 𝑆 𝑔 βˆ— -open sets of ( π‘Œ , 𝜎 ). Since 𝑓 is 𝑆 𝑔 βˆ— -irresolute, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since 𝑓 is onto, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡) are nonempty. Now 𝑋 = 𝑓 βˆ’1 π‘Œ = 𝑓 βˆ’1 𝐴 βˆͺ 𝐡 = 𝑓 βˆ’1 𝐴 βˆͺ 𝑓 βˆ’1 ( 𝐡). Thus 𝑋 is
  • 4.
    𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -ConnectedSpaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 20| the union of disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . This contradicts the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected and hence ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Theorem 4.11: If a surjective map 𝑓 : 𝑋, 𝜏 β†’ π‘Œ , 𝜎 is strongly 𝑆 𝑔 βˆ— -continuous and 𝑋, 𝜏 is a connected space, then ( π‘Œ , 𝜎 ) is 𝑆 𝑔 βˆ— -connected. Proof: Similar to the proof of the above theorem. Theorem 4.12: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a perfectly 𝑆 𝑔 βˆ— -continuous map, 𝑋, 𝜏 a connected space, then ( π‘Œ , 𝜎 ) has an indiscrete topology. Proof: Suppose that there exists a proper open set U of ( π‘Œ , 𝜎 ), then U is 𝑆 𝑔 βˆ— -open in ( π‘Œ , 𝜎 ). Since 𝑓 is perfectly 𝑆 𝑔 βˆ— -continuous, 𝑓 βˆ’1 ( π‘ˆ) is a proper open and closed subset of 𝑋, 𝜏 . This implies 𝑋, 𝜏 is not connected which is a contradiction. Therefore ( π‘Œ , 𝜎 ) has an indiscrete topology. Theorem 4.13: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a totally 𝑆 𝑔 βˆ— -continuous map, from a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto any space ( π‘Œ , 𝜎 ), then ( π‘Œ , 𝜎 ) is an indiscrete space. Proof: Suppose ( π‘Œ , 𝜎 ) is not indiscrete. Let A be a proper nonempty open subset of ( π‘Œ , 𝜎 ). Then 𝑓 βˆ’1 ( 𝐴) is a proper nonempty 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed subset of 𝑋, 𝜏 , which is a contradiction to the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Then ( π‘Œ , 𝜎 ) must be indiscrete. Theorem 4.14: If 𝑓 is a contra 𝑆 𝑔 βˆ— -continuous map from a 𝑆 𝑔 βˆ— -connected space 𝑋, 𝜏 onto any space ( π‘Œ , 𝜎 ), then ( π‘Œ , 𝜎 ) is not a discrete space. Proof: Suppose ( π‘Œ , 𝜎 ) is discrete. Let A be any proper nonempty open and closed subset of ( π‘Œ , 𝜎 ). Then 𝑓 βˆ’1 ( 𝐴) is a proper nonempty 𝑆 𝑔 βˆ— -open and 𝑆 𝑔 βˆ— -closed subset of 𝑋, 𝜏 , which is a contradiction to the fact that 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Hence (π‘Œ, 𝜎 ) is not a discrete space. Theorem 4.15: Suppose that 𝑋 is a 𝑆 𝑔 βˆ— - 𝑇 1 2 space then 𝑋 is connected if and only if it is 𝑆 𝑔 βˆ— -connected. Proof: Suppose that 𝑋 is connected. Then 𝑋 cannot be written as a union of two non-empty disjoint proper subsets of X. Suppose 𝑋 is not 𝑆 𝑔 βˆ— -connected. Let 𝐴 and 𝐡 be any two 𝑆 𝑔 βˆ— -open sets subsets of 𝑋 such that 𝑋 = 𝐴 βˆͺ 𝐡, where 𝐴 ∩ 𝐡 = βˆ…. Since 𝑋 is a 𝑆 𝑔 βˆ— - 𝑇 1 2 space, every 𝑆 𝑔 βˆ— -open sets are open. Hence 𝐴 and 𝐡 are open sets which contradicts the fact that 𝑋 is not connected. Then 𝑋 is 𝑆 𝑔 βˆ— -connected. The converse part follows from the theorem that every 𝑆 𝑔 βˆ— -connected space is connected. Theorem 4.16: If 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is slightly 𝑆 𝑔 βˆ— -continuous surjective function and X is 𝑆 𝑔 βˆ— -connected then Y is connected. Proof: Suppose Y is not connected. Then there exists non-empty disjoint open set A and B such that Y= 𝐴 βˆͺ 𝐡. Therefore A and B are clopen sets in Y. Since 𝑓 is slightly 𝑆 𝑔 βˆ— -continuous and surjective, 𝑓 βˆ’1 ( 𝐴) and 𝑓 βˆ’1 ( 𝐡)are non-empty disjoint 𝑆 𝑔 βˆ— -opensets in X. Also 𝑓 βˆ’1 π‘Œ = 𝑋 = 𝑓 βˆ’1 𝐡 . This shows that X is not 𝑆 𝑔 βˆ— - connected, a contradiction. Hence Y is connected. Theorem 4.17: If 𝑓 is slightly 𝑆 𝑔 βˆ— -continuous function from a connected space ( 𝑋, 𝜏 ) onto a space ( π‘Œ , 𝜎 ) then Y is not a discrete space. Proof: Suppose that Y is a discrete space. Let A be a proper nonempty open subset of Y. Then 𝑓 βˆ’1 ( 𝐴) is nonempty 𝑆 𝑔 βˆ— -clopen subset of X, which is a contradiction to the fact that X is 𝑆 𝑔 βˆ— -connected. Hence Y is not a discrete space. Theorem 4.18: A space X is 𝑆 𝑔 βˆ— -connected if every slightly 𝑆 𝑔 βˆ— -continuous from X into any 𝑇 π‘œ space Y is constant. Proof: Let every slightly 𝑆 𝑔 βˆ— -continuous function from a space X into Y be constant. If X is not 𝑆 𝑔 βˆ— -connected then there exists a proper nonempty 𝑆 𝑔 βˆ— -clopen subset A of X . Let ( π‘Œ , 𝜎 ) be such that Y= π‘Ž , 𝑏 𝜎 = {βˆ…, 𝑦 , π‘Ž , 𝑏 } be a topology. Let 𝑓 : 𝑋 β†’ π‘Œ be any function such that 𝑓 𝐴 = { π‘Ž } and 𝑓 𝑋 βˆ’ 𝐴 = 𝑏 . Then f is a non-constant and slightly 𝑆 𝑔 βˆ— -continuous function such that Y is 𝑇 π‘œ which is a contradiction. Hence Y is 𝑆 𝑔 βˆ— -connected. Theorem 4.19: A space 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected if and only if every totally 𝑆 𝑔 βˆ— -continuous function from a space 𝑋, 𝜏 into any 𝑇 0 space ( π‘Œ , 𝜎 ) is a constant map. Proof: Suppose 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) is a totally 𝑆 𝑔 βˆ— -continuous function where ( π‘Œ , 𝜎 ) is a 𝑇 0-space. Suppose that f is not a constant map, then we can select two points π‘₯ and 𝑦 such that 𝑓 π‘₯ β‰  𝑓 𝑦 . Since ( π‘Œ , 𝜎 ) is a 𝑇 0-spaceand 𝑓 ( π‘₯ ) and 𝑓 ( 𝑦 ) are distinct points of Y,there exists an open set G in ( π‘Œ , 𝜎 ) containing 𝑓 ( π‘₯ ) but not 𝑓 ( 𝑦 ). Since 𝑓 is a totally 𝑆 𝑔 βˆ— -continuous function, 𝑓 βˆ’1 ( 𝐺) is a 𝑆 𝑔 βˆ— -clopen subset of π‘₯ , 𝜏 . Clearly π‘₯ ∈ 𝑓 βˆ’1 ( 𝐺) and 𝑦 βˆ‰ 𝑓 βˆ’1 𝐺 . Now 𝑋 = 𝑓 βˆ’1 𝐺 βˆͺ 𝑓 βˆ’1 𝐺 c which is the union of non-empty 𝑆 𝑔 βˆ— -open subsets of X. Thus X is not 𝑆 𝑔 βˆ— -connected space,which contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Hence f is a constant map. Conversely, suppose ( 𝑋, 𝜏 ) is not a 𝑆 𝑔 βˆ— -connected space there exists a proper non-empty 𝑆 𝑔 βˆ— -clopen subset 𝐴 of 𝑋. Let π‘Œ = { π‘Ž , 𝑏 } and 𝜏 = π‘Œ , βˆ…, π‘Ž , 𝑏 be a topology for π‘Œ . Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a function
  • 5.
    𝑆𝑔 βˆ— -Compact and 𝑆𝑔 βˆ— -ConnectedSpaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 5 | Iss.2| Feb. 2015 | 21| such that 𝑓 𝐴 = { π‘Ž } and 𝑓 π‘Œ 𝐴 = 𝑏 . Then 𝑓 is non-constant and totally 𝑆 𝑔 βˆ— -continuous such that π‘Œ is 𝑇 0, which is a contradiction. Hence X must be 𝑆 𝑔 βˆ— -connected. Theorem 4.20: Let 𝑓 : 𝑋, 𝜏 β†’ ( π‘Œ , 𝜎 ) be a totally 𝑆 𝑔 βˆ— -continuous function & π‘Œ is a 𝑇 1-space. If A is a non- empty 𝑆 𝑔 βˆ— -connected subset of X. Then 𝑓 ( 𝐴)is singleton. Proof: Suppose that 𝑓 ( 𝐴)is not a singleton . Let 𝑓 ( π‘₯ 1 ) = 𝑦 1 ∈ 𝐴 and 𝑓 ( π‘₯ 2 ) = 𝑦 2 ∈ 𝐴. Since 𝑦 1, 𝑦 2 ∈ π‘Œ and Y is a 𝑇 1-space, there exists an open set G in ( π‘Œ , 𝜎 ) containing y,but not 𝑦 2. Since 𝑓 is totally 𝑆 𝑔 βˆ— - continuous, 𝑓 βˆ’1 ( 𝐺) is 𝑆 𝑔 βˆ— -continuous, 𝑓 βˆ’1 ( 𝐺) is 𝑆 𝑔 βˆ— -clopen set containing π‘₯ 1 but not π‘₯ 2. Now 𝑋 = 𝑓 βˆ’1 𝐺 βˆͺ 𝑓 βˆ’1 𝐺 c . Thus we have expressed X as a union of two non-empty 𝑆 𝑔 βˆ— -open sets. This contradicts the fact that X is 𝑆 𝑔 βˆ— -connected. Therefore 𝑓 ( 𝐴) is singleton. Theorem 4.21: Every semi-connected space is 𝑆 𝑔 βˆ— -connected. Proof: Let 𝑋, 𝜏 be a semi-connected space. Suppose that 𝑋, 𝜏 is not 𝑆 𝑔 βˆ— -connected, then 𝑋 = 𝐴 βˆͺ 𝐡 where A and B are disjoint nonempty 𝑆 𝑔 βˆ— -open sets in 𝑋, 𝜏 . Since every 𝑆 𝑔 βˆ— -open set is semi-open, 𝑋, 𝜏 is not a semi-connected space which is a contradiction and hence 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected. Remark 4.22: The converse of theorem 4.21 is not true as can be seen from the following example. Example 4.23: Let 𝑋 = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } and 𝜏 = { 𝑋, πœ™, π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑏 , 𝑏 , 𝑐 , π‘Ž , 𝑐 , { π‘Ž , 𝑏 , 𝑐 }}. Then 𝑆𝑂 𝑋, 𝜏 = { 𝑋, πœ™, π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑏 , 𝑏 , 𝑐 , π‘Ž , 𝑐 , π‘Ž , 𝑑 , 𝑏 , 𝑑 , 𝑐 , 𝑑 , π‘Ž , 𝑏 , 𝑐 , π‘Ž , 𝑐 , 𝑑 , π‘Ž , 𝑏 , 𝑑 , 𝑏 , 𝑐 , 𝑑 } and 𝑆 𝑔 βˆ— 𝑂 𝑋, 𝜏 = 𝜏 Here 𝑋, 𝜏 is 𝑆 𝑔 βˆ— -connected but not semi-connected because 𝑋 = π‘Ž βˆͺ { 𝑏 , 𝑐 , 𝑑 } where π‘Ž and { 𝑏 , 𝑐 , 𝑑 } are non-empty disjoint semi-open sets. REFERENCES [1] Das.P., I.J.M.M. 12,1974,31-34. [2] Dorsett.C., Semi compactness, semi separation axioms and product spaces,Bull. Of Malaysian Mathematical sciences Society, 4(1), 1981, 21-28. [3] Pious Missier.S and Arul Jesti.J. , A new notion of open sets in topological spaces, Int.J.Math.Arc., 3(11), 2012, 3990-3996. [4] Pious Missier.S and Arul Jesti.J., Some new functions in topological spaces, Outreach(A multi-disciplinary research Journal), 7, 2014,161-168. [5] Pious Missier.S and Arul Jesti.J., Properties of 𝑆 𝑔 βˆ— -functions in topological spaces, Math. Sci. Int. Research Journal(IMRF), 3(2), 2014, 911-915. [6] Pious Missier.S and Arul Jesti.J., 𝑆 𝑔 βˆ— -homeomorphism in topological spaces, Outreach(A multi-disciplinary research Journal) (Communicated). [7] Pious Missier.S and Arul Jesti.J., Slightly 𝑆 𝑔 βˆ— -continuous functions and Totally 𝑆 𝑔 βˆ— -continuous functions in Topological spaces,IOSR-JM, (Accepted). [8] Willard S., General topology, Addison Wesley, 1970.