GENERALIZED LOCALLY CLOSED SETS IN IDEAL ...

BULLETIN OF THE ALLAHABAD MATHEMATICAL SOCIETY GOLDEN JUBILEE YEAR VOLUME Vol. ??, Part ?, 200?, ????.

GENERALIZED LOCALLY CLOSED SETS IN IDEAL TOPOLOGICAL SPACES M. Navaneethakrishnan and D. Sivaraj (Received 27 October 2007; Revised 3 July 2008; Re-revised 5 July 2008) We define and characterize I-locally ?-closed and Ig -locally ?-closed sets and discuss their properties.

1. Introduction Local function in topological spaces using ideals was introduced by Kuratowski [6]. More importance was given to the topic, ideal topological spaces, by Vaidyanathaswamy [10]. In 1990, Jankovic and Hamlett [5] gave a brief account of all results established earlier and established some new results in ideal topological spaces. In [2], a new class of sets called Ig -closed sets were defined and discussed. More characterizations and properties of Ig -closed sets are given in [8]. In this paper, in secction 3, we define and study a new class of generalized locally closed sets in ideal topological spaces. In section 4, we define and study a new class of generalized closed sets in ideal topological spaces. 2. Preliminaries Let (X, τ ) be a topological space. An ideal I [6] is defined as a non-empty collection of subsets of X which satisfying the following two conditions: (1) A ∈ I and B ⊂ A, then B ∈ I; (2) A ∈ I and B ∈ I, then A ∪ B ∈ I. An ideal topological space is a topological space (X, τ ) with an ideal I on X and is denoted by (X, τ, I). For a subset A of X, A? (I) = {x ∈ X | U ∩ A 6∈ I for each neighborhood U of x } is called the local function of A with respect to I and τ [6]. When there is no chance for confusion, we will simply write A? for A? (I). X ? is often a proper subset of X. The hypothesis X = X ? [4] is equivalent to the hypothesis τ ∩ I = {∅} [9]. For every ideal topological space (X, τ, I), there exists a topology τ ? , 2000 Mathematics Subject Classification: 54A05. Key words and phrases: Ig -closed, Ig -open, locally closed, I-locally closed and Idense sets. c 2008 Allahabad Mathematical Society

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M. Navaneethakrishnan and D. Sivaraj

finer than τ, generated by β(I, τ ) = {U − I | U ∈ τ and I ∈ I} but in general, β(I, τ ) is not always a topology [5]. A subset A of an ideal space (X, τ, I) is τ ? -closed or ?-closed [5] (resp.?-dense in itself [4], ?-perfect [4], I-dense [3]) if A? ⊂ A (resp.A ⊂ A? , A = A? , A? = X). A subset A of an ideal space (X, τ, I) is Ig -closed [2] if A? ⊂ U whenever U is open and A ⊂ U. A subset A of an ideal space (X, τ, I) is said to be Ig -open if X − A is Ig -closed. A subset A of an ideal space (X, τ, I) is said to be I-locally closed [1] if A = U ∩ V where U is open and V is ?-perfect. Equivalently, A is I-locally closed if and only if A = U ∩ A? for some open set U. By a space, we always mean a topological space (X, τ ) with no separation properties assumed. If A ⊂ X, cl(A) and int(A) will, respectively, denote the closure and interior of A in (X, τ ) and cl? (A) and int? (A) will, respectively, denote the closure and interior of A in (X, τ ? ). A subset A of a space (X, τ ) is said to be g-closed [7] if cl(A) ⊂ U whenever A ⊂ U and U is open. 3. I-locally?-closed sets. In this section, we define and study a new class of generalized locally closed sets in an ideal topological space (X, τ, I). A subset A of an ideal space (X, τ, I) is said to be I-locally ?-closed if there exist an open set U and a ?-closed set F such that A = U ∩ F. If I = {∅}, then I-locally ?closed sets coincide with locally closed sets. The following Theorem 3.1 gives characterizations of I-locally?-closed sets. THEOREM 3.1. Let (X, τ, I) be an ideal space and A be a subset of X. Then the following are equivalent. (a) A is I-locally ?-closed. (b) A = U ∩ cl? (A) for some open set U. (c) A? − A is closed. (d) (X − A? ) ∪ A = A ∪ (X − cl? (A)) is open. (e) A ⊂ int(A ∪ (X − A? )). PROOF.(a)⇒(b). If A is I-locally ?-closed, then there exist an open set U and a ?-closed set F such that A = U ∩ F. Clearly, A ⊂ U ∩ cl? (A). SinceF is ?- closed, cl? (A) ⊂ cl? (F ) = F and so U ∩ cl? (A) ⊂ U ∩ F = A. Therefore, A = U ∩ cl? (A). A?

(b)⇒(c). Now A? − A = A? ∩ (X − A) = A? ∩ (X − (U ∩ cl? (A)) = ∩ (X − U ). Therefore,A? − A is closed.

GENERALIZED LOCALLY CLOSED SETS IN IDEAL TOPOLOGICAL SPACES

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(c)⇒(d). Since X − (A? − A) = (X − A? ) ∪ A, (X − A? ) ∪ A is open. Clearly, (X − A? ) ∪ A = A ∪ (X − cl? (A)). (d)⇒(e) is clear. (e)⇒(a). X − A? = int(X − A? ) ⊂ int(A ∪ (X − A? )) which implies that A ∪ (X − A? ) ⊂ int(A ∪ (X − A? )) and so A ∪ (X − A? ) is open. Since A = (A ∪ (X − A? )) ∩ cl? (A), A is I-locally?-closed Every open subset of an ideal space (X, τ, I) is always I-locally ?closed, since X is ?-closed. The following Example 3.2 shows the reverse direction is not true. Theorem 3.3 below shows that the two concepts coincide for I-dense sets. EXAMPLE 3.2. Let X be any nonempty set with a non-discrete topology. If I = ℘(X), then every subset of X is ?-closed and hence Ilocally ?-closed. So there exist I-locally ?-closed sets which are not open. THEOREM 3.3. Let (X, τ, I) be an ideal space and A be a subset of X. If A is I-locally ?-closed and I-dense, then A is open. PROOF. If A is I-locally ?- closed, by Theorem 3.1(e), A ⊂ int(A ∪ (X − A? )). Since A is I-dense, A? = X and so A ⊂ int(A) which implies that A is open. COROLLARY 3.4. Let (X, τ, I) be an ideal space and A be an I-dense subset of X. Then, A is I- locally ?-closed if and only if A is open. Clearly, every ?-closed set is I-locally?-closed. The following Example 3.5 shows that the converse need not be true. Theorem 3.6 below shows that the reverse direction is true for Ig -closed sets. Also, Theorem 3.6 is a generalization of Theorem 2.3 of [2] which shows that every Ig -closed, open set is ?-closed. EXAMPLE 3.5. Consider the ideal space (X, τ, I) where X = {a, b, c, d}, τ = {∅, {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}, X} and I = {∅, {a}, {c}, {a, c}}. If A = {b}, then A is I- locally ?- closed, since A is open and X is ?closed. Since A? = {a, b, c, d}, A is not ?- closed. THEOREM 3.6. Let (X, τ, I) be an ideal space and A be an Ig -closed subset of X. Then, A is I-locally ?-closed if and only if A is ?- closed. PROOF. If A is ?-closed, then A is I-locally ?-closed. Conversely, suppose A is I-locally ?-closed and Ig -closed. By Theorem 2.12 of [2], A? − A has no nonempty closed set. By Theorem 3.1(c), A? − A is closed. Therefore, A? − A = ∅ which implies that A? ⊂ A and so A is ?-closed.

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THEOREM 3.7. Let (X, τ ) be a space and A be a g-closed subset of X. Then, A is locally closed if and only if A is closed. PROOF. If I = {∅}, then Ig -closed sets coincide with g-closed sets [2, Theorem 2.2(1)] and τ ? = τ [5]. Therefore, the proof follows from Theorem 3.6, if we take I = {∅}. In any ideal space (X, τ, I), locally closed sets are I-locally ?-closed sets, since closed sets are ?-closed sets. The following Example 3.8 shows that the reverse implication is not true. Theorem 3.9 below shows that for ?-dense in itself sets, the three collection of sets, namely, I-locally ?-closed sets, I-locally closed sets and locally closed sets coincide. EXAMPLE 3.8. Let X = {a, b, c, d} with the topology τ = {∅, {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}, X}. If I = ℘(X), then every subset of X is ?closed. If A = {b, c, d}, then A is ?-closed and so A is I-locally?-closed. Since X is the only open set containing A and A is not closed, A is not locally closed. THEOREM 3.9. Let (X, τ, I) be an ideal space and A be a ?-dense in itself subset of X. Then the following are equivalent. (a) A is I-locally closed. (b) A is locally closed. (c) A is I-locally?-closed. PROOF. (a) ⇒ (b) and (b) ⇒ (c) are clear. (c)⇒(a). If A is I-locally?-closed, by Theorem 3.1(b), A = U ∩cl? (A) for some open set U. Since A is ?-dense in itself, cl? (A) = A? and so A = U ∩ A? which implies that A is I-locally closed. An ideal space (X, τ, I) is said to be a TI -space [2] if every Ig -closed subset of X is a ?-closed set. The following Theorem 3.10 gives a characterization of TI -spaces. THEOREM 3.10. (X, τ, I).

The following are equivalent in an ideal space

(a) (X, τ, I) is a TI -space. (b) Every Ig -closed set is an I-locally?-closed set. PROOF. (a)⇒(b). If A is Ig -closed, then by hypothesis, A is ?-closed and so A is I-locally ?-closed.


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(b)⇒(a). If A is Ig -closed, then A? − A contains no nonempty closed set [2, Theorem 2.12]. By (b), A is I-locally ?-closed and so A? −A is closed by Theorem 3.1(c). Hence, A? − A = ∅ which implies that A is ?-closed. Therefore, (X, τ, I) is a TI -space. 4. Ig -locally ?-closed sets In this section, we define and study a new class of generalized closed sets in an ideal topological space (X, τ, I). DEFINITION 4.1. A subset A of an ideal space (X, τ, I) is said to be Ig -locally ?-closed if there exist an Ig -open set U and a ?-closed set F such that A = U ∩ F. Since every open set is Ig -open, every I-locally ?-closed set is a Ig locally ?-closed set. The following Example 4.2 shows that the converse is not true. Theorem 4.3 below gives characterizations of Ig -locally ?- closed sets. EXAMPLE 4.2. Let X = {a, b, c, d} with the topology τ = {∅, {b}, {a, b}, {b, c}, {a, b, c}, {a, b, d}, X} and I = {∅, {a}, {c}, {a, c}}. If A = {a}, then A ∈ I and so A? = ∅ which implies that A is Ig -closed. If B = X − A, then B is Ig -open and so B is Ig -locally ?-closed. Since X is the only open set containing B and B is not ?-closed, B is not I-locally ?-closed. THEOREM 4.3. Let (X, τ, I) be an ideal space and A be a subset of X. Then the following are equivalent. (a) Ig -locally ?-closed. (b) A = U ∩ cl? (A) for some Ig -open set U. (c) cl? (A) − A = A? − A is Ig -closed. (d) A ∪ (X − cl? (A)) = A ∪ (X − A? ) is Ig -open. PROOF. (a)⇒(b). If A is Ig - locally ?-closed, then there exist an Ig -open set U and a ?-closed set F such that A = U ∩ F. Clearly, A ⊂ U ∩ cl? (A). Since F is ?-closed, cl? (A) ⊂ cl? (F ) = F and so U ∩ cl? (A) ⊂ U ∩ F = A. Therefore, A = U ∩ cl? (A). (b)⇒(c). Now cl? (A) − A = A? − A = A? ∩ (X − A) = A? ∩ (X − (U ∩ = A? ∩(X −U ). Let V be an open set such that cl? (A)−A ⊂ V. Then A? ∩(X−U ) ⊂ V and so (X−U ) ⊂ (X−A? )∪V. Since X−U is Ig -closed and (X −A? )∪V is open, cl? (X −U ) ⊂ (X −A? )∪V and so A? ∩ cl? (X −U ) ⊂ V. Since A? ∩(X−U ) ⊂ A? , (A? ∩(X−U ))? ⊂ (A? )? ⊂ A? . Also, A? ∩(X−U ) ⊂ cl? (A))

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X − U implies that (A? ∩ (X − U ))? ⊂ (X − U )? ⊂ cl? (X − U ). Therefore, (A? ∩ (X − U ))? ⊂ A? ∩ cl? (X − U ) ⊂ V. Hence, (cl? (A) − A)? ⊂ V and so cl? (A) − A is Ig -closed. (c)⇒(d). Since X − (cl? (A) − A) = A ∪ (X − cl? (A)), A ∪ (X − cl? (A)) is Ig -open. (d)⇒(a) Since A = (A ∪ (X − cl? (A))) ∩ cl? (A) and cl? (A) is ?-closed, by hypothesis, A is Ig -locally ?-closed. If G is an open subset of an ideal space (X, τ, I), then clearly G is Ig locally ?-closed. The following Example 4.4 shows that the converse is not true. Theorem 4.5 below shows that the two concepts coincide for I-dense subsets. Theorem 4.8 below gives another characterization of Ig -locally ?-closed sets. EXAMPLE 4.4. Consider the ideal space in Example 4.1. If U = {b, c, d}, then U is Ig -open. If F = {a, d}, then F ? = {d} and so F is ?-closed. Therefore, {d} = U ∩ F is Ig -locally ?-closed but {d} is not open. THEOREM 4.5. Let (X, τ, I) be an ideal space and A be a subset of X. If A is Ig -locally ?-closed and I-dense, then A is Ig -open. PROOF. If A is Ig -locally ?-closed, by Theorem 4.3(d), A ∪ (X − cl? (A)) is Ig -open. Since A is I-dense, A? = X and so cl? (A) = X which implies that A is Ig -open. COROLLARY 4.6. Let (X, τ, I) be an ideal space and A be an I-dense subset of X. Then, A is Ig -locally ?-closed if and only if A is Ig -open. COROLLARY 4.7. Let (X, τ, I) be an ideal space. Then the following are equivalent. (a) Every subset of X is Ig -locally ?-closed. (b) Every τ ? -dense set is Ig -open. PROOF.(a)⇒(b) follows from Theorem 4.3(d). (b)⇒(a). For any subset A of X, consider F = A ∪ (X − cl? (A)). Then cl? (F ) = cl? (A ∪ (X − cl? (A))) = X and so F is τ ? -dense. By hypothesis, F is Ig -open. By Theorem 4.3, A is Ig -locally ?-closed. THEOREM 4.8. Let (X, τ, I) be an ideal space and A be a subset of X. Then, A is Ig -locally ?-closed if and only if A ∩ (A? − A)? is Ig -open. PROOF. If A is any subset of X, then (cl? (A) ∩ (X − A))? − (cl? (A) ∩ (X − A)) = (cl? (A) ∩ (X − A))? ∩ (X − (cl? (A) ∩ (X − A))) = (cl? (A) ∩


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(X − A))? ∩ ((X − cl? (A)) ∪ A) = ((cl? (A) ∩ (X − A))? ∩ (X − cl? (A))) ∪ ((cl? (A) ∩ (X − A))? ∩ A) = ∅ ∪ ((cl? (A) − A)? ∩ A) = (A? − A)? ∩ A. Now A is Ig -locally ?-closed if and only if cl? (A) − A is Ig -closed, by Theorem 4.3(c) if and only if cl? (A) ∩ (X − A) is Ig -closed if and only if (cl? (A) ∩ (X − A))? − (cl? (A) ∩ (X − A)) is Ig -open, by Theorem 2.12 of [8] if and only if (A? − A)? ∩ A is Ig -open. Acknowledgement The authors sincerely thank the referee for the valuable suggestions. References [1] J. Dontchev, Idealization of Ganster Reilly Decomposition Theorems, Math. GN /9901017, 5, Jan. 1999(Internet). [2] J. Dontchev, M. Ganster and T. Noiri, Unified approach of generalized closed sets via topological ideals, Math. Japonica, 49(1999), 395-401. [3] J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology and its Applications, 93(1999), 1-16. [4] E. Hayashi, Topologies defined by local properties, Math.Ann., 156(1964), 205-215. [5] D. Jankovic and T.R.Hamlett, New Topologies from old via Ideals, Amer. Math. Monthly, 97(4)(1990), 295-310. [6] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. [7] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo 19(2)(1970), 89-96. [8] M. Navaneethekrishnan and J. Paulraj Joseph, g-Closed sets in ideal topological spaces, Acta Math. Hungar.,(to appear) [9] P. Samuels, A topology formed from a given topology and ideal, J. London Math. Soc., 10(2)(1975), 409-416. [10] R. Vaidyanathaswamy, Set Topology, Chelsea Publishing Company, 1946.

Department of Mathematics Kamaraj College, Thoothukudi Tamil Nadu, India. E-mail: [email protected] Department of Computer Applications D.J. Academy for Managerial Excellence Coimbatore-641 032, Tamil Nadu, India. E-mail: ttn− [email protected]