J. Comp. & Math. Sci. Vol.4 (6), 431-436 (2013)
rpsI-Connectedness and Compactness in Ideal Topological Spaces B. SAKTHI SATHYA1 and S. MURUGESAN2 1
Department of Mathematics, G. Venkataswamy Naidu College Kovilpatti, Tamilnadu, INDIA. 2 Department of Mathematics, S. Ramasamy Naidu Memorial College Sattur, Tamilnadu, INDIA. (Received on: November 10, 2013) ABSTRACT In this paper, we introduce and study the notion of rpsI connectedness and compactness in ideal topological spaces. Keywords: rpsI-connected, pgprI-connected. AMS Subject Classification: 54A05, 54A10.
1. INTRODUCTION Ideals in topological spaces have been considered since 1930. This topic has won its importance by the paper of Applications to Vaidyanathaswamy12. various fields were further investigated by Jankovic and Hamlett4, Dontchev et al.2, Mukherjee et al.7, Arenas et al.1, Navaneetha Krishnan et al.9, Nasef and Mahmoud8, etc. The purpose of this paper is to introduce and study the notion of rpsI-connectedness in ideal topological spaces. Throughout the present paper, (X,τ) or (Y,σ) will denote a topological space with no separation properties assumed. For a subset A of a topological space (X,τ), cl(A)
and int(A) will denote the closure and interior of A in (X,τ), respectively. A topological space X is said to be connected if X cannot be written as the union of two disjoint non empty open sets. An ideal is defined as a non-empty collection I of subsets of X satisfying the following two conditions. i. If A ∈ I and B ⊆ A, then B ∈ I ii. If A ∈ I and B ∈ I then A ∪ B ∈ I Given a topological space (X, τ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (.) : ℘(X) → ℘(X) is called a local function[6] of A with respect to τ and I is defined as follows: for A
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⊆ X, A*(I,τ) = {x ∈ X / A ∩ U ∉ I, for every U ∈ τ(x)}, where τ(x) = {U ∈ τ / x ∈ U}. A Kuratowski closure operator cl*(.) for a topology τ*(I,τ), called the * topology finer than τ, is defined by cl*(A) = A ∪ A*(I,τ)3. When there is no chance for confusion we will simply write A* for A*(I,τ) and τ* or τ*( I), for τ*(I,τ). If A is an ideal on X, then (X,τ,I) is called an ideal topological space or simply an ideal space. 2. PRELIMINARIES Definition 2.1 A subset A of an ideal topological space (X, τ, I) is called regular I-open5 if A = int(cl*(A)) Definition 2.2 A subset A of an ideal topological space (X, τ, I) is called i. regular generalized I-closed10(rgIclosed) if cl*(A) ⊆ U whenever A ⊆ U and U is regular I-open. ii. pre generalized pre regular I-closed10 (pgprI-closed) if pIcl(A) ⊆ U whenever A ⊆ U and U is regular generalized Iopen. iii. regular pre semi I-closed10(rpsI-closed) if spIcl(A) ⊆ U whenever A ⊆ U and U is regular generalized I-open. The complement of the above mentioned generalized I-closed sets are their respective I-open sets. The following lemma will be useful in sequel. Definition 2.4 A function f:(X,τ,I) → (Y,σ) is called rpsI-continuous11 if f -1(A) is rpsIclosed in X, for every closed subset A of Y.
Definition 2.5 A function f:(X,τ,I) → (Y,σ) is called rpsI-irresolute11 if f -1(A) is rpsIclosed in X, for every rpsI-closed subset A of Y. Lemma 2.610 Every pgprI-open set is rpsI-open. 3. rpsI-Connectedness In this section we introduce and study the concept of rpsI-connected and also discuss about some of their properties. Definition 3.1 An ideal topological space (X,τ,I) is said to be rpsI-connected if X cannot be written as the disjoint union of two non empty rpsI-open sets. If X is not rpsIconnected it is said to be rpsI-disconnected. Theorem 3.2 Let (X,τ,I) be an ideal topological space. If X is rpsI-connected, then X cannot be written as the union of two disjoint non empty rpsI-closed sets. Proof: Suppose not, that is X = A∪B where A and B are rpsI-closed sets, A ≠ Φ, B ≠ Φ and A∩B = Φ. Then A = Bc and B = Ac. Since A and B are rpsI-closed which implies that A and B are rpsI-open sets. Therefore X is not rpsI-connected which is a contradiction. Hence the proof. Theorem 3.3 For an ideal topological space (X,τ,I), the following are equivalent
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B. Sakthi Sathya, et al., J. Comp. & Math. Sci. Vol.4 (6), 431-436 (2013)
i) X is rpsI-connected ii) X and Φ are the only subsets of X which are both rpsI-open and rpsI-closed. Proof is obvious. Remark 3.4 The subspace of a rpsI-connected space need not be a rpsI-connected. For example, consider the ideal topological space (X,τ,I), where X = {a,b,c,d} with τ = {Φ, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {Φ, {a}}. In this ideal space, {b,c,d} is rpsI-connected but the subset {b,c} is not rpsI-connected. Remark 3.5
If A and C are rpsI-connected subset of an ideal topological space (X,τ,I) and if A⊆B⊆C, then B is not rpsIconnected. For example consider the ideal space in Remark 3.4, let A={b}, B={b,c}, C={b,c,d}. In this ideal space A and C are rpsI-connected but B is not rpsI-connected. Remark 3.6 The intersection of two rpsIconnected space need not be a rpsIconnected. For example consider the ideal topological space (X,τ,I), where X = {a,b,c,d} with τ = {Φ, X, {a}, {b,d}, {a,b,d}} and I = {Φ, {a}}. In this ideal space, {c,d} and {b,c} are rpsI-connected but their intersection{c} is not rpsIconnected. Theorem 3.7 Let f:(X,τ,I) → (Y,σ) be a function. If X is rpsI-connected and f is rpsI-irresolute, surjective, then Y is rpsI-connected.
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Proof: Suppose that Y is not rpsIconnected. Let Y = A ∪ B, where A and B are disjoint non-empty rpsI-open sets in Y. Since f is rpsI-irresolute and onto f-1(Y) = f-1(A ∪ B) which implies X = f-1(A) ∪ f1(B) and f-1(A) ∩ f-1(B) = f-1(A ∩ B) = f1(Φ) = Φ, where f-1(A) and f-1(B) are disjoint nonempty rpsI-open sets in X. This contradicts to the fact that X is rpsIconnected. Hence Y is rpsI-connected. Theorem 3.8 Let f:(X,τ,I) → (Y,σ) be a function. If X is rpsI-connected and f is rpsI-continuous, surjective, then Y is connected. Proof is similar to that of Theorem 3.7. Theorem 3.9 Every rpsI-connected space is connected. Proof: Let X be rpsI-connected. Suppose X is not connected, then there exists a proper non-empty subset B of X which is both open and closed in X. Since every closed set is rpsI-closed, B is a proper nonempty subset of X which is both rpsI-open and rpsI-closed in X. Then by Theorem 3.3, X is not rpsI-connected. This proves the theorem. The converse of the above theorem need not be true as seen from the following example. Consider the ideal
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topological space in Remark 3.4, the set {b,c} is connected but not rpsI-connected. Definition 3.10 An ideal topological space (X,τ,I) is said to be pgprI-connected if X cannot be written as the disjoint union of two non empty pgprI-open sets. Theorem 3.11
Example 3.16 Consider the ideal topological space in Remark 3.4, the set {a,b,c} is connected but not semiI-connected. The concepts of semiI-connected and pgprI-connected are independent of each other. For example, consider the ideal topological space in Remak 3.4, the set {b,c} is semiI-connected but not pgprIconnected. Also the set {a,b,d} is pgprIconnected but not semiI-connected.
For an ideal topological space (X,τ,I), the following are equivalent i) X is pgprI-connected ii) X and Φ are the only subsets of X which are both pgprI-open and pgprI-closed. Proof is obvious.
From Theorem 3.9, Theorem 3.12, Example 3.13, Remark 3.15, Example 3.16 we have the following diagram Connected semiI-connected rpsIconnected pgprI-connected
Theorem 3.12
4. rpsI-COMPACTNESS
Every rpsI-connected space is pgprIconnected. The converse need not be true as seen from the following example.
Definition 4.1
Example 3.13 Consider the ideal topological space in Remark 3.4, the set {a,b,d} is pgprIconnected but not rpsI-connected. Definition 3.14 An ideal topological space (X,τ,I) is said to be semiI-connected if X cannot be written as the disjoint union of two non empty semiI-open sets. Remark 3.15 Every semiI-connected space is connected but the converse need not be true.
A collection {Aα : α ∈ ∇} of rpsIopen sets in a topological space X is called a rpsI-open cover of a subset B of X if B ⊆ ∪{ Aα : α ∈ ∇} holds. Definition 4.2 An ideal topological space (X, τ,I) is said to be rpsI-compact if every rpsI-open cover of X has a finite subcover. Definition 4.3 A subset B of an ideal topological space (X,τ,I) is said to be rpsI-compact relative to X if for every collection {Aα : α ∈ ∇} of rpsI-open subsets of X such that B ⊆ ∪{ Aα : α ∈ ∇}, there exists a finite subset ∇0 of ∇ such that B ⊆ {Aα : α ∈ ∇0}.
Journal of Computer and Mathematical Sciences Vol. 4, Issue 6, 31 December, 2013 Pages (403-459)
B. Sakthi Sathya, et al., J. Comp. & Math. Sci. Vol.4 (6), 431-436 (2013)
Theorem 4.4 i)
A rpsI-continuous image of a rpsIcompact space is compact. ii) If a map f:(X,τ,I) → (Y,σ) is rpsIirresolute and a subset B of X, then f(B) is rpsI-compact relative to Y. Proof: i)
Let f:(X,τ,I) → (Y,σ) be a rpsIcontinuous map from a rpsI-compact space X onto a topological space Y. Let {vα:α∈ ∇} be an open cover of Y. Then {f-1(vα):α∈ ∇} is a rpsI-open cover of X. Since X is rpsI-compact, it has a finite subcover say{ f-1(v1), f-1(v2),…., f-1(vn)}. Since f is onto{v1,v2,…..,vn} is a cover of Y and so Y is compact. ii) Let { vα:α∈ ∇} be any collection of rpsIopen subsets of Y such that f(B) ⊆ ∪ {vα:α∈ ∇}. Then B ⊆ ∪ {f-1(vα):α∈ ∇} holds. By hypothesis, there exists a finite subset ∇0 of ∇ such that B ⊆ ∪ {f1(vα):α∈ ∇0}. Therefore we have f(B) ⊆ ∪ {vα:α∈ ∇0} which shows that f(B) is rpsI-compact relative to Y. Definition 4.5 An ideal topological space (X,τ,I) is said to be semiI-compact[5] if every semiIopen cover of X has a finite subcover. Theorem 4.6 Every semiI-compact is rpsI-compact. Proof is obvious. Theorem 4.7 Every rpsI-closed subset of a compact space is compact.
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Proof: Since every closed set is rpsI-closed and closed subset of a compact space is compact. REFERENCES 1. F.G. Arenas, J.Dontchev, M.L.Puertas, Idealization of some weak separation axioms, Acta Math. Hungar.,89(1-2),4753(2000). 2. J. Dontchev, M. Ganster, D. Rose, Ideal Resolvability, Topology and Its Applications, 93,1-16 (1993). 3. Jamal M. Mustafa, Contra SemiIcontinuous functions, Hacettepe Journal of Mathematics and Statistics, Vol. 39(2),191-196 (2010). 4. D. Jankovic and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97,295-310 (1990). 5. A.Keskin and T. Noiri, S. Yuksel, Idealization of a decomposition theorem, Math. Hungar.,102,269-277(2004). 6. K. Kuratowski, Topology, Vol I, Academic Press New York, (1966). 7. M.N. Mukherjee, R. Bishwambhar, R. Sen, On extension of topological spaces in terms of Ideals, Topology and its Applications, 154,3167-3172 (2007). 8. A. A. Nasef, R.A. Mahmoud, Some applications via fuzzy ideals, Chaos, Solitons and Fractols, 13,825-831 (2002). 9. M. Navaneethakrishnan, I. Paulraj Joseph, g-closed sets in ideal topological spaces, Acta Math. Hungar., DOI: 10.107/S10474-007-7050-1. 10. B.Sakthi alias Sathya, S.Murugesan, On regular pre semiI-closed sets in ideal topological spaces, International Journal of Mathematics and soft computing, Vol. 39(1) 37-46 (2013).
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