EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 299-314 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Between closed and Ig-closed sets Néstor Raúl Pachón Rubiano Departamento de Matemáticas, Escuela Colombiana de Ingeniería, Bogotá, Colombia. Departamento de Matemáticas, Universidad Nacional, Bogotá, Colombia. Abstract. The concept of closed sets is a central object in general topology. In order to extend many of important properties of closed sets to a larger families, Norman Levine initiated the study of generalized closed sets. In this paper we introduce, via ideals, new generalizations of closed subsets, which are strong forms of the Ig-closed sets, called ρIg-closed sets and closed-I sets. We present some properties and applications of these new sets and compare the ρIg-closed sets and the closed-I sets with the g-closed sets introduced by Levine. We show that I-closed and closed-I are independent concepts, as well as I∗-closed sets and closed-I concepts. 2010 Mathematics Subject Classifications: 54D30, 54C10 Key Words and Phrases: g-closed, Ig-closed, I-compact, I-normal, I-QHC, ρC(I)-compact. 1. Introduction and preliminaries The g-closed sets, which is a extension of closed sets, was introduced by Levine and the Ig-closed sets, which is a generalization of g-closed sets, was defined by Jafari-Rajesh, in terms of ideals. In this paper we introduce and study new intermediate concepts between closed and Ig-closed sets, via ideals. We also present some applications of these new sets, related to compactness and normality. An ideal I in a set X is a subset of P(X), the power set of X, such that: (i) if A ⊆ B ⊆ X and B ∈ I then A ∈ I, and (ii) if A ∈ I and B ∈ I then A ∪B ∈ I. Some simple and useful ideals in X are: (i) P(A), where A ⊆ X, (ii) If (X), the ideal of all finite subsets of X, and (iii) Ic (X), the ideal of all countable subsets of X. Email addresses: nestor.pachon@escuelaing.edu.co, nrpachonr@unal.edu.co (N.R. Pachón) http://www.ejpam.com 299 © 2018 EJPAM All rights reserved. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 300 If (X, τ) is a topological space and I is an ideal in X, then (X, τ, I) is called an ideal space. If (X, τ) is a topological space and A ⊆ X then the closure and the interior of A are denoted by A (or adhτ (A)) and 0 A (or intτ (A)), respectively. If A and B are subsets of the space (X, τ) and A ∩B = ∅ = A ∩B then A and B are called separated. If A ⊆ 0 A then A is said to be pre-open [6]. If 0 A ⊆ A then A is defined to be pre-closed [6]. It is clear that A is pre-open if and only if X\A is pre-closed. If (X, τ) is a topological space and A ⊆ X then A is said to be g-closed [5] if, for each U ∈ τ , A ⊆ U implies A ⊆ U . An ideal space (X, τ, I) is defined to be I-normal [1] if for every pair of disjoint closed subsets F and G, there exist disjoint open sets U and V such that F\U ∈ I and G\V ∈ I. The symbol □ is used to indicate the end of a proof. 2. ρIg-closed sets The generalized closed sets via ideals, that we consider, are due to Jafari-Rajesh and these are extensions of the g-closed sets of Levine. In this section we define the ρIg- closed sets, which is a new intermediate concept between closed and Ig-closed sets. Some properties, characterizations and applications are presented. If (X, τ, I) is an ideal space and A ⊆ X then A is defined to be Ig-closed [2] if, for all U ∈ τ , A ⊆ U implies A\U ∈ I. It is noted that closed → g-closed → Ig-closed. Definition 2.1. If (X, τ, I) is an ideal topological space and A ⊆ X then A is said to be ρIg-closed if for each U ∈ τ , if A\U ∈ I then A\U ∈ I. It is clear that closed → ρIg-closed → Ig-closed The converse are not true, as we can see in the next example. Example 2.2. (1) If U is the usual topology in the set R, then all A ⊆ R is ρIg-closed in the ideal space (R,U , I = P(R)), but (0, 1) is not g-closed. Then ρIg-closed↛g-closed and so ρIg-closed↛closed. (2) If C = {∅,R}∪{(r,∞) : r ∈ R} then Z is not ρIg-closed in the space (R, C, I = Ic (R)), because Z\ (0,∞) ∈ I but Z\ (0,∞) = (−∞, 0] /∈ I. However Z is g-closed, and then Z is Ig-closed. Thus g-closed↛ ρIg-closed and Ig-closed ↛ ρIg-closed. Observe that g-closed and ρIg-closed are independent concepts. An application of ρIg-closed sets is shown in the next theorem. If (X, τ, I) is an ideal space then a subset A is said to be: (1) I-compact [8] if for each open cover {Vα}α∈Λ of A, there exists Λ0 ⊆ Λ, finite, such that A\ ∪ α∈Λ0 Vα ∈ I, and N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 301 (2) ρI-compact [9] if for each family {Vα}α∈Λ of open subsets of X, if A\ ∪ α∈Λ Vα ∈ I there exists Λ0 ⊆ Λ, finite, such that A\ ∪ α∈Λ0 Vα ∈ I. The space (X, τ, I) is I-compact if X is I-compact, and (X, τ, I) is ρI-compact if X is ρI-compact. Theorem 2.3. If the ideal space (X, τ, I) is ρI-compact and A ⊆ X we have that: (1) If A is closed then A is ρI-compact. (2) If A is ρIg-closed then A is ρI-compact. (3) If A is Ig-closed then A is I-compact. Proof. (1) Let {Vα}α∈Λ be a collection of open sets of X such that A\ ∪ α∈Λ Vα ∈ I, this is, X\ [ (X\A) ∪ ∪ α∈Λ Vα ] ∈ I. There exists Λ0 ⊆ Λ, finite, with X\ [ (X\A) ∪ ∪ α∈Λ0 Vα ] ∈ I, this is, A\ ∪ α∈Λ0 Vα ∈ I. (2) Let {Vα}α∈Λ be a collection of open sets of X such that A\ ∪ α∈Λ Vα ∈ I. Since A is ρIg-closed we have that A\ ∪ α∈Λ Vα ∈ I. Given that A is ρI-compact, there exists Λ0 ⊆ Λ, finite, with A\ ∪ α∈Λ0 Vα ∈ I. Hence A\ ∪ α∈Λ0 Vα ∈ I. (3) Let {Vα}α∈Λ be a collection of open sets of X such that A ⊆ ∪ α∈Λ Vα. Given that A is Ig-closed we have that A\ ∪ α∈Λ Vα ∈ I. But A is ρI-compact, and so there exists Λ0 ⊆ Λ, finite, with A\ ∪ α∈Λ0 Vα ∈ I. Thus A\ ∪ α∈Λ0 Vα ∈ I. □ We recall that an ideal space (X, τ, I) is said to be Ig-normal if for every pair of disjoint g-closed subsets F and G of X, there exist disjoint open sets U and V such that F\U ∈ I and G\V ∈ I. Renukadevi-Sivaraj have shown that if (X, τ, I) is I-compact and (X, τ) is T2 then (X, τ, I) is I-normal. In contrast we have the following result. Theorem 2.4. If (X, τ, I) is ρI-compact and (X, τ) is T2 then (X, τ, I) is Ig-normal. Proof. Suppose that F and G are disjoint g-closed sets. It is noted that, by Theorem 2.3, F and G are I-compact subsets of X. Let g ∈ G, arbitrary. For each f ∈ F there are disjoint Uf ∈ τ and Vf ∈ τ such that f ∈ Uf and g ∈ Vf . Given that F ⊆ ∪ f∈F Uf and N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 302 F is I-compact, there exists F0 ⊆ F , finite, with F\ ∪ f∈F0 Uf ∈ I. Let Tg = ∪ f∈F0 Uf and Wg = ∩ f∈F0 Vf . It is noted that Tg ∩Wg = ∅ and F\Tg ∈ I. Now, since G ⊆ ∪ g∈G Wg and G is I-compact, there exists G0 ⊆ G, finite, with G\ ∪ g∈G0 Wg ∈ I. If V = ∪ g∈G0 Wg and U = ∩ g∈G0 Tg then U and V are disjoint, G\V ∈ I and F\U =∪ g∈G0 (X\Tg) ∈ I. □ If I is an ideal in X and B ⊆ X, it is easy to see that the set IB = {I ∩B : I ∈ I} is an ideal in B. Theorem 2.5. Let (X, τ, I) be an ideal space. If A ⊆ X and B ⊆ X then: (1) If A and B are ρIg-closed then A ∪B is ρIg-closed. (2) A is ρIg-closed if and only if, for each closed set F , if F\ ( A\A ) ∈ I then F ∈ I. (3) If A\B ∈ I, B\A ∈ I and A is ρIg-closed then B is ρIg-closed. (4) If A ⊆ B ⊆ A and A is ρIg-closed, then B is ρIg-closed. (5) If A is ρIg-closed and B is closed, then A ∩B is ρIg-closed. (6) If A ⊆ B and A is ρIg-closed in the space (X, τ, I), then A is ρ(IB)g-closed in the space (B, τB, IB), where τB = {U ∩B : U ∈ τ}. Proof. (1) Suppose that U ∈ τ and (A ∪B) \U ∈ I. Then A\U ∈ I and B\U ∈ I, and so A\U ∈ I and B\U ∈ I. This implies that A ∪B\U ∈ I. (2) (→) Suppose that A is ρIg-closed, F ⊆ X is closed and that F\ ( A\A ) ∈ I, this is, F ∩ [( X\A ) ∪A ] ∈ I. Then F ∩ ( X\A ) ∈ I and A\ (X\F ) = F ∩ A ∈ I. Since A is ρIg-closed we have that A\ (X\F ) ∈ I, this is F ∩ A ∈ I. Thus F =( F ∩A ) ∪ [ F ∩ ( X\A )] ∈ I. (←) Let U ∈ τ with A\U ∈ I. Given that A\U = ( A\U ) \ ( A\A ) and A\U is closed, the hypothesis implies that A\U ∈ I. (3) Suppose that V ∈ τ and B\V ∈ I. Since A\V ⊆ (A\B) ∪ (B\V ) ∈ I then A\V ∈ I. Given that A is ρIg-closed we have that A\V ∈ I. Hence ( A\V ) ∪ ( B\A ) ∈ I. But B\V ⊆ ( A\V ) ∪ ( B\A ) and so B\V ∈ I. (4) It is a consequence of (3). N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 303 (5) If U ∈ τ and (A ∩B) \U ∈ I, this is, A\ [U ∪ (X\B)] ∈ I, then A\ [U ∪ (X\B)] ∈ I because A is ρIg-closed. Thus ( A ∩B ) \U ∈ I. Now, A ∩B\U ⊆ ( A ∩B ) \U =( A ∩B ) \U and so A ∩B\U ∈ I. (6) Suppose that V ∈ τB and A\V = I0 ∈ IB. There are U ∈ τ and I ∈ I with V = B ∩ U and I0 = I ∩ B. Then A\V = A\ (B ∩ U) = B ∩ I and this implies that A\U ⊆ A\ (B ∩ U) = B ∩ I ⊆ I. Thus A\U ∈ I. Since A is ρIg-closed we have that A\U ∈ I. This implies that ( A\U ) ∩ B ∈ IB, this is, ( A ∩B ) \U ∈ IB, and finally adhτB (A) \V = adhτB (A) \ (U ∩B) = ( B ∩A ) \ (U ∩B) = ( B ∩A ) \U ∈ IB. □ Example 2.6. Let C = {∅,R} ∪ {(r,∞) : r ∈ R} , I = If (R), A = 2Z and B = {p ∈ Z : |p| is a prime number}. We have that, in the space (R, C, I), A and B are ρIg-closed sets, because if U ∈ C and A\U ∈ I (or B\U ∈ I) then U = R and so A\U ∈ I ( or B\U ∈ I ) . However A ∩ B is not ρIg-closed since A ∩ B = {−2, 2}, (A ∩B) \ (0,∞) ∈ I, but A ∩B\ (0,∞) = (−∞, 0] /∈ I. The following result is due to Newcomb. Lemma 2.7. If f : X → Y is a function we have that: (1) If I is an ideal in X, then f(I) = {f(I) : I ∈ I} is an ideal in Y . (2) If f is inyective and J is an ideal in Y , then the set f−1 (J ) = { f−1(J) : J ∈ J } is an ideal in X. Theorem 2.8. (1) If f : (X, τ) → (Y, β) is a continuous, closed and inyective function, I is an ideal on X, J = f(I) and if A ⊆ X is ρIg-closed, then f(A) is ρJg-closed. (2) If f : (X, τ)→ (Y, β) is a continuous, closed and inyective function, I is an ideal on X, J = { V ⊆ Y : f−1 (V ) ∈ I } and if A ⊆ X is ρIg-closed, then f(A) is ρJg-closed. (3) If f : (X, τ)→ (Y, β) is a continuous, open and inyective function, I is an ideal on X, J = { V ⊆ Y : f−1 (V ) ∈ I } and if B ⊆ Y is ρJg-closed, then f−1(B) is ρIg-closed. (4) If f : (X, τ)→ (Y, β) is an inyective, continuous and closed function, J is an ideal in Y and if A is ρ ( f−1(J ) ) g -closed, then f(A) is ρJg-closed. Proof. (1) If V ∈ β and f(A)\V ∈ J then A\f−1(V ) ∈ I, because f is inyective. Given that A is ρIg-closed we have that A\f−1(V ) ∈ I, and so f [ A\f−1(V ) ] ∈ f(I). But f ( A ) \V ⊆ f ( A ) \f(f−1(V )) ⊆ f [A\f−1(V )]. Moreover f(A) ⊆ f(A), since f is closed. In consequence f(A)\V ∈ f(I). (2) It is similar to (1). N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 304 (3) If U ∈ τ and f−1(B)\U ∈ I, this is, f−1 [B\f(U)] ∈ I, then B\f(U) ∈ J . Given that B is ρJg-closed we have that B\f(U) ∈ J . In consequence f−1 [ B\f(U) ] ∈ I, this is f−1 ( B ) \U ∈ I. Since f is continuous we have that f−1(B) ⊆ f−1(B), and so f−1 (B)\U ∈ I. (4) If W ∈ β and f(A)\W ∈ J then A\f−1 (W ) = f−1 [f (A) \W ] ∈ f−1 (J ). Since A is ρ ( f−1(J ) ) g -closed there is J ∈ J with A\f−1 (W ) = f−1 (J). But f (A)\W ⊆ f ( A ) \W ⊆ f ( A ) \f [ f−1 (W ) ] ⊆ f [ A\f−1 (W ) ] = f ( f−1 (J) ) ⊆ J , and so f (A)\W ∈ J . □ Definition 2.9. If (X, τ, I) is an ideal topological space and A ⊆ X then A is said to be ρIg-open if X\A is ρIg-closed. The following result is a consequence of Theorem 2.5. Theorem 2.10. Let (X, τ, I) be an ideal space. If A ⊆ X and B ⊆ X then: (1) If A and B are ρIg-open then A ∩B is ρIg-open. (2) A is ρIg-open if and only if, for each closed set F , if F\ ( A\ 0 A ) ∈ I then F ∈ I. (3) If B\A ∈ I, 0 A\ 0 B ∈ I and A is ρIg-open then B is ρIg-open. (4) If 0 A ⊆ B ⊆ A and A is ρIg-open, then B is ρIg-open. (5) If A is ρIg-open and B is open, then A ∪B is ρIg-open. Next we present other useful properties of ρIg-open sets. Theorem 2.11. If (X, τ, I) is an ideal space then A ⊆ X is ρIg-open if and only if, for each F ⊆ X, closed, if F\A ∈ I then F\ 0 A ∈ I. Proof. (→) Suppose that F ⊆ X is closed and that F\A ∈ I, this is, (X\A) \ (X\F ) ∈ I. Given that X\A is ρIg-closed we have that X\A\ (X\F ) ∈ I or, equivalently, F\ 0 A ∈ I. (←) Suppose that V ∈ τ and (X\A) \V ∈ I, this is, (X\V ) \A ∈ I. The hypothesis implies that (X\V ) \ 0 A ∈ I, or equivalently, ( X\ 0 A ) \V ∈ I. Hence X\A\V ∈ I and so X\A is ρIg-closed. □ Theorem 2.12. If (X, τ, I) is an ideal space, then A ⊆ X is ρIg-closed if and only if A\A is ρIg-open. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 305 Proof. (→) Suppose that F ⊆ X is closed and that F\ ( A\A ) ∈ I. By the Theorem 2.5 we have that F ∈ I, and so F\int ( A\A ) ∈ I, because int ( A\A ) = ∅. Thus A\A is ρIg-open. (←) Suppose that U ∈ τ and that A\U ∈ I. Given that ( A\U ) \ ( A\A ) = A\U ∈ I and A\A is ρIg-open, the Theorem 2.11 implies ( A\U ) \int ( A\A ) ∈ I, this is, A\U ∈ I. □ Theorem 2.13. If A and B are ρIg-open subsets of an ideal space (X, τ, I), such that A ∩B ∈ I and A ∩B ∈ I, then A ∪B is ρIg-open. Proof. Suppose that F ⊆ X is closed and that F\ (A ∪B) ∈ I. We have that: (a) F\A ∪B ∈ I. (b) ( F ∩A ) \A ∈ I, because ( F ∩A ) \A ⊆ ( A ∩B ) ∪ [F\ (A ∪B)] ∈ I. (c) ( F ∩B ) \B ∈ I. (d) ( F ∩A ) \ 0 A ∈ I, because F ∩A is closed and A is ρIg-open. (e) ( F ∩B ) \ 0 B ∈ I. (f) [ F ∩A ∪B ] \ ( 0 A ∪ 0 B ) ∈ I. In fact, given that[( F ∩A ) \ 0 A ] ∪ [( F ∩B ) \ 0 B ] ∈ I and ( A ∪B ) \ ( 0 A ∪ 0 B ) ⊆ ( A\ 0 A ) ∪ ( B\ 0 B ) , we have that[ F ∩A ∪B ] \ ( 0 A ∪ 0 B ) = F ∩ [( A ∪B ) \ ( 0 A ∪ 0 B )] ⊆ F ∩ [( A\ 0 A ) ∪ ( B\ 0 B )] =[( F ∩A ) \ 0 A ] ∪ [( F ∩B ) \ 0 B ] , and so [ F ∩A ∪B ] \ ( 0 A ∪ 0 B ) ∈ I. (g) F\ 0 (A ∪B) ∈ I, because F\ 0 (A ∪B) ⊆ F\ ( 0 A ∪ 0 B ) ⊆ [( F ∩A ∪B ) \ ( 0 A ∪ 0 B )] ∪ ( F\A ∪B ) ∈ I. Therefore A ∪B is ρIg-open. □ Corollary 2.14. (1) If A and B are separated ρIg-open subsets of an ideal space (X, τ, I) then A ∪B is ρIg-open. (2) If A and B are ρIg-closed subsets of an ideal space (X, τ, I), such that X\ ( 0 A ∪B ) ∈ I and X\ ( A ∪ 0 B ) ∈ I, then A ∩B is ρIg-closed. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 306 We end this section with an application to ρIg-open sets to I-normality. Theorem 2.15. The ideal space (X, τ, I) is I-normal if and only if, for each pair of disjoint closed sets F and G, there are disjoint ρIg-open sets A and B such that F\A ∈ I and G\B ∈ I. Proof. (→) This is simple because open → ρIg-open. (←) If F and G are disjoint closed sets, then there exist disjoint ρIg-open sets A and B with F\A ∈ I and G\B ∈ I. The Theorem 2.11 implies that F\ 0 A ∈ I and G\ 0 B ∈ I. Moreover 0 A and 0 B are disjoint open sets. □ 3. Closed-I sets In this section we introduce the closed-I sets, an intermediate concept between closed sets and ρIg-closed sets. We also consider some applications of these sets. Given an ideal space (X, τ, I) and a set A ⊆ X, we denote by A∗ (I) = {x ∈ X : U ∩A /∈ I, for every U ∈ τ with x ∈ U} , written simply as A∗ when there is no chance for confusion. It is clear that A∗ ⊆ A. A Kuratowski closure operator for a topology τ∗ (I), finer than τ , is defined by Cl∗ (A) = A∪A∗, for all A ⊆ X. When there is no chance for confusion τ∗ (I) is denoted by τ∗. The topology τ∗ has as a base β (τ, I) = {V \I : V ∈ τ and I ∈ I} [12] . In 1990, D. Jancovic and T. R. Hamlett introduced the notion of I-open sets. If (X, τ, I) is an ideal space and A ⊆ X, A is said to be I-open [3] if A ⊆ int (A∗). A is said to be I-closed if X\A is I-open. In 1992, D. Jancovic and T. R. Hamlett introduced the notion of I∗-open sets. If (X, τ, I) is an ideal space and A ⊆ X, A is said to be I∗-closed [4] if A∗ ⊆ A or, equivalently, if A is closed in (X, τ∗). A is said to be I∗-open if X\A is I∗-closed. Definition 3.1. If (X, τ, I) is an ideal space and A ⊆ X, then A is said to be closed-I if A\A ∈ I. A subset B is defined to be open-I if X\B is closed-I. It is observed that: (1) closed → closed-I. (2) A is open-I if and only if A\ 0 A ∈ I. (3) A is closed-I and open-I if and only if Fr(A) ∈ I, where Fr(A) is the frontier of A. (4) A is closed-I if and only if A\A is open-I. (5) Each I ∈ I is open-I. (6) If A is open then A is open-I. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 307 Example 3.2. (1) If U is the usual topology in R and if I = If (R), then [0, 1) is closed-I but [0, 1) is not closed. Since Q\Q /∈ I then Q is not closed-I. (2) If X = {a, b, c, d}, τ = {∅, X, {c} , {a, b} , {a, b, c}} and I = {∅, {a}}, then the set A = {b, c, d} is I-open [7]. Now, since A\ 0 A = {b, d} /∈ I then A is not open-I. It is noted that {a, c} /∈ τ but {a, c} is open-I. Moreover, since A\A = {a} ∈ I and A∗ = {a, b, d} ⊈ A then A is closed-I but A is not I∗-closed. Now, if B = {a} then B\B = {b, d} /∈ I and B∗ = ∅ ⊆ B. Then B is I∗-closed but B is not closed-I. (3) If I = {∅, {c} , {d} , {c, d}} and τ = {∅, X, {d} , {a, c} , {a, c, d}}, where X = {a, b, c, d}, then the set A = {a, c, d} is open-I. However A is not I-open [7]. Hence, in general, open ↛ I-open. In consequence the I-closed and closed-I are independent concepts, as well as I∗-closed and closed-I concepts. Theorem 3.3. Let (X, τ, I) be an ideal space. If A ⊆ X and B ⊆ X then: (1) If A is closed-I then A is ρIg-closed. (2) If A is closed-I then ( A )∗ \A ∈ I, and so ( 0 A )∗ \A ∈ I. (3) If A and B are closed-I then A ∪B and A ∩B are closed-I. (4) If A\B ∈ I, B\A ∈ I and A is closed-I then B is closed-I. (5) If A ⊆ B ⊆ A and A is closed-I, then B is closed-I. (6) If A is pre-open and closed-I then A is open-I. (7) If A ⊆ B and A is closed-I in (X, τ, I), then A is closed-IB in (B, τB, IB). (8) If A ⊆ B, A is closed-IB in (B, τB, IB), B is closed-I in (X, τ, I), then A is closed-I in (X, τ, I). Proof. (1) Suppose that U ∈ τ and A\U ∈ I. Given that A\U ⊆ ( A\A ) ∪ (A\U) ∈ I, we have that A\U ∈ I. (2) Since A is closed then ( A )∗ ⊆ A, and so ( 0 A )∗ \A ⊆ ( A )∗ \A ⊆ A\A ∈ I. (3) It is enough to note that A ∪B\ (A ∪B) = ( A ∪B ) \ (A ∪B) ⊆ ( A\A ) ∪ ( B\B ) ∈ I, and that A ∩B\ (A ∩B) ⊆ ( A ∩B ) \ (A ∩B) ⊆ ( A\A ) ∪ ( B\B ) ∈ I. (4) Since B\B ⊆ ( A\A ) ∪ ( B\A ) ∪ (A\B) ∈ I, we have that B\B ∈ I. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 308 (5) It is a consequence of (4). (6) By hypothesis, A\ 0 A ⊆ A\A ∈ I. (7) Given that adhτB (A) \A = ( A ∩B ) \A = ( A\A ) ∩B ∈ IB, then adhτB (A) \A ∈ IB. (8) We have that B\B ∈ I and adhτB (A) \A ∈ IB ⊆ I. Now, adhτB (A) = A ∩ B and A\A ⊆ [( A ∩B ) \A ] ∪ ( B\B ) ∈ I. □ Example 3.4. (1) In the space (R, C, I) of Example 2.2, Z is g-closed but Z is not closed- I, because Z is not ρIg-closed. (2) If C = {∅,R} ∪ {(r,∞) : r ∈ R} and I = P ((0,∞)), then the set A = (−∞, 0) is ρIg-closed in the space (R, C, I) because if U ∈ C and A\U ∈ I then U = R, and so A\U ∈ I. However, since A\A = {0} /∈ I, we have that A is not closed-I. Thus, in general, ρIg-closed ↛closed-I. (3) If U is the usual topology in R and I = P ({0, 1}), then the set A = (0, 1) is not g-closed. However, given that A\A = {0, 1} ∈ I, we conclude that (0, 1) is closed-I. So, in general, closed-I ↛g-closed. Thus, closed-I and g-closed are independent concepts. We have the following diagram. g − Closed Closed Closed− I ρIg − Closed Ig − Closed In the Theorem 3.5 we review the behavior of closed-I sets under continuous or closed functions. Theorem 3.5. (1) If (Y, β,J ) is an ideal space, f : (X, τ)→ (Y, β) is a continuous and inyective function and B is closed-J , then f−1 (B) is closed-f−1 (J ). (2) If (X, τ, I) is an ideal space, f : (X, τ)→ (Y, β) is a closed function and A is closed-I, then f (A) is closed-f(I). (3) If (X, τ, I) is an ideal space, f : (X, τ) → (Y, β) is a continuous function, J ={ D ⊆ Y : f−1 (D) ∈ I } and B is closed-J , then f−1 (B) is closed-I. (4) If f : (X, τ)→ (Y, β) is an inyective and closed function, J is an ideal in Y and if A is closed-f−1(J ), then f(A) is closed-J . N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 309 Proof. (1) We have that f−1 (B)\f−1 (B) ⊆ f−1 ( B ) \f−1(B) = f−1 ( B\B ) ∈ f−1(J ), given that B\B ∈ J . (2) Since f(A)\f(A) ⊆ f ( A ) \f (A) ⊆ f ( A\A ) ∈ f (I), then f(A)\f(A) ∈ f (I). (3) Given that B\B ∈ J then f−1 ( B ) \f−1 (B) = f−1 ( B\B ) ∈ I. But f−1 (B)\f−1 (B) ⊆ f−1 ( B ) \f−1 (B) and so f−1 (B)\f−1 (B) ∈ I. (4) There is J ∈ J such that A\A = f−1 (J), and so f (A)\f (A) ⊆ f ( A ) \f (A) ⊆ f ( A\A ) = f ( f−1 (J) ) ⊆ J . Hence f (A)\f (A) ∈ J . □ The following theorem is a consequence of Theorem 3.3. Theorem 3.6. Let (X, τ, I) be an ideal space. If A ⊆ X and B ⊆ X then: (1) If A is open-I then A is ρIg-open. (2) If A and B are open-I then A ∪B and A ∩B are open-I. (3) If B\A ∈ I, 0 A\ 0 B ∈ I and A is open-I then B is open-I. (4) If 0 A ⊆ B ⊆ A and A is open-I, then B is open-I. (5) If A is pre-closed and open-I then 0 A is closed-I. (6) If A ⊆ B and A is open-I in (X, τ, I), then A is open-IB in (B, τB, IB). Some applications of the closed-I and open-I sets are shown now. A subset A of an ideal space (X, τ, I) is said to be σI-compact [9] if for each nonempty collection {Vα}α∈Λ of nonempty open sets, if A\ ∪ α∈Λ Vα ∈ I then there exists Λ0 ⊆ Λ, finite, such that A ⊆ ∪ α∈Λ0 Vα. The space (X, τ, I) is σI-compact if X is σI-compact. It is simple to see that if (X, τ, I) is σI-compact and if A ⊆ X is closed then A is σI-compact. Theorem 3.7. If (X, τ, I) is an ideal space and A ⊆ X is closed-I we have that: (1) If (X, τ, I) is ρI-compact then A is ρI-compact. (2) If (X, τ, I) is σI-compact then A is σI-compact. Proof. (1) It is a consequence of Theorem 2.3, because closed-I → ρIg-closed. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 310 (2) Let {Vα}α∈Λ be a nonempty collection of nonempty open sets with A\ ∪ α∈Λ Vα ∈ I. Since A\A ∈ I and A\ ∪ α∈Λ Vα ⊆ ( A\ ∪ α∈Λ Vα ) ∪ ( A\A ) ∈ I then A\ ∪ α∈Λ Vα ∈ I. But A is σI-compact and so there exists Λ0 ⊆ Λ, finite, such that A ⊆ A ⊆ ∪ α∈Λ0 Vα. □ Theorem 3.8. The ideal space (X, τ, I) is I-normal if and only if, for each pair of disjoint closed sets F and G, there are disjoint open-I sets A and B such that F\A ∈ I and G\B ∈ I. Proof. (→) It is clear because open→open-I. (←) It is a consequence of Theorem 2.15 since open-I → ρIg-open. □ Remark 3.9. If (X, τ, I) is an ideal space and A ⊆ X, then: (1) τ ⊕ I is the topology generated for the base τ ∪ I. It is noted that τ ⊕ I = { V ∪ ∪ C : V ∈ τ and C ⊆ P(I) } . (2) I(A) is the set ∪ I∈I, I⊆A I. In the next Theorem 3.10 we show that τ ⊕ I is the smallest topology in X, that contains τ , such that all open-I set is an open set. Theorem 3.10. If (X, τ, I) is an ideal space we have that: (1) If A ⊆ X then intτ⊕I (A) = intτ (A) ∪ I(A). (2) A set F ⊆ X is closed in the space (X, τ ⊕ I) if and only if there exists G ⊆ X, closed in (X, τ), and a collection C ⊆ P(I), such that F = G\ ∪ C. (3) If A ⊆ X then adhτ⊕I (A) = adhτ (A) \I(X\A). (4) τ ⊕ I is the smallest topology β in X such that: (a) τ ⊆ β and (b) In the space (X,β, I), for each A ⊆ X, A is open-I if and only if A is open. Proof. (1) It is clear that intτ (A)∪I(A) ∈ τ ⊕I and that intτ (A)∪I(A) ⊆ A, and so intτ (A)∪ I(A) ⊆ intτ⊕I (A). Now, suppose that W ∈ τ ⊕ I and that W ⊆ A. There exist V ∈ τ and a collection {Iα}α∈Λ of elements in I, such that W = V ∪ ∪ α∈Λ Iα. Since V ⊆ A then V ⊆ intτ (A). Given that, for all α ∈ Λ, Iα ⊆ A then ∪ α∈Λ Iα ⊆ I(A), and so W ⊆ intτ (A) ∪ I(A). In particular intτ⊕I (A) ⊆ intτ (A) ∪ I(A). N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 311 (2) It is obvious. (3) Given that A ⊆ adhτ (A) \I(X\A) and adhτ (A) \I(X\A) is closed in (X, τ ⊕ I) then adhτ⊕I (A) ⊆ adhτ (A) \I(X\A). Now, suppose that F is closed in (X, τ ⊕ I) and that A ⊆ F . There exists G ⊆ X, closed in (X, τ), and a collection C ⊆ P(I), such that F = G\ ∪ C. Since adhτ (A) ⊆ G and ∪ C ⊆ I (X\A) we have that adhτ (A) \I (X\A) ⊆ G\ ∪ C = F . In particular, adhτ (A) \I (X\A) ⊆ adhτ⊕I (A). (4) (i) Suppose that B ⊆ X is open-I in the space (X, τ ⊕ I, I), this is B\intτ⊕I (B) ∈ I⊆τ ⊕ I. Since B = [B\intτ⊕I (B)] ∪ intτ⊕I (B) then B ∈ τ ⊕ I. (ii) Suppose that β is a topology in X such that τ ⊆ β and that in the space (X,β, I), for each A ⊆ X, A is open-I if and only if A is open. Given that all I ∈ I is open-I in (X,β, I) then, by hypothesis, I ⊆ β. Hence τ⊕I⊆β. □ Remark 3.11. If I is an ideal in X and J is an ideal in Y , then I ⊗ J is the set of all D ⊆ X × Y such that there exist I ∈ I, A ⊆ X, J ∈ J and B ⊆ Y , with D ⊆ (A× J) ∪ (I ×B). Theorem 3.12. (1) If I is an ideal in X and J is an ideal in Y , then I ⊗ J is an ideal in X × Y . (2) If A is open-I in the space (X, τ, I) and B is open-J in the space (Y, β,J ), then A×B is open-I ⊗ J in the space (X × Y, τ × β, I ⊗ J ). Proof. (1) It is clear that if V ⊆ W ⊆ X × Y and W ∈ I ⊗ J , then V ∈ I ⊗ J . Sup- pose that {D1, D2} ⊆ I ⊗ J . There are {I1, I2} ⊆ I, {J1, J2} ⊆ J , {A1, A2} ⊆ P (X) and {B1, B2} ⊆ P (Y ) such that D1 ⊆ (A1 × J1) ∪ (I1 ×B1) and D2 ⊆ (A2 × J2)∪(I2 ×B2). Hence D1∪D2 ⊆ (A1 × J1)∪(A2 × J2)∪(I1 ×B1)∪(I2 ×B2) ⊆ [(A1 ∪A2)× (J1 ∪ J2)] ∪ [(I1 ∪ I2)× (B1 ∪B2)]. This implies that D1 ∪D2 ∈ I ⊗ J . (2) Since A\ 0 A ∈ I and B\ 0 B ∈ J , we have that (A×B) \int (A×B) = (A×B) \ ( 0 A× 0 B ) =[( A\ 0 A ) ×B ] ∪ [ A× ( B\ 0 B )] ∈ I ⊗ J . □ 4. Other characteristics of the topology τ ⊕ I In this section we present some properties of the topology τ ⊕I, related to normality, compactness and C-compactness. N.R. Pachón / Eur. J. Pure Appl. Math, 11 (1) (2018), 299-314 312 Remark 4.1. If (X, τ, I) is an ideal space then I⊛ = {∪ C : C ⊆ P(I) } and I = { J : J ⊆ I, for some I ∈ I } It is clear that I⊛ = P(UI), where UI = ∪ I∈I I. It is easy to see that I is an ideal in X, that I ⊆ I⊛, I ⊆ I, and that if I ∈ I then I ∈ I. Moreover, if τ is a topology in X, it is clear that τ ⊕ I = τ ⊕ I⊛. Theorem 4.2. If I is an ideal in X, τ is a topology in X and (X, τ ⊕ I) is a normal space, then (X, τ, I⊛) is I⊛-normal. Proof. Suppose that F and G are disjoint closed sets in (X, τ). Since F and G are closed sets in (X, τ ⊕ I), there exists disjoint sets U ∪ ∪ α∈Λ1 Iα ∈ τ ⊕ I and V ∪ ∪ α∈Λ2 Iα ∈ τ ⊕ I such that F ⊆ U ∪ ∪ α∈Λ1 Iα and G ⊆ V ∪ ∪ α∈Λ2 Iα. Thus F\U ⊆ ∪ α∈Λ1 Iα ∈ I⊛ and G\V ⊆ ∪ α∈Λ2 Iα ∈ I⊛. Moreover U and V are disjoint open sets in (X, τ). □ A space (X, τ) is said to be: (1) QHC [11] if for each open cover {Vα}α∈Λ of X, there exists Λ0 ⊆ Λ, finite, such that X = ∪ α∈Λ0 Vα. (2) C-compact [13] if for each closed set F and each open cover {Vα}α∈Λ of F , there exists Λ0 ⊆ Λ, finite, such that F ⊆ ∪ α∈Λ0 Vα. An ideal space (X, τ, I) is defined to be: (1) ρI-QHC [10] if for each collection {Vα}α∈Λ of open sets, if X\ ∪ α∈Λ Vα ∈ I there exists Λ0 ⊆ Λ, finite, such that X\ ∪ α∈Λ0 Vα ∈ I. (2) ρC(I)-compact [10] if for each closed set F and each collection {Vα}α∈Λ of open sets, if F\ ∪ α∈Λ Vα ∈ I there exists Λ0 ⊆ Λ, finite, such that F\ ∪ α∈Λ0 Vα ∈ I. Theorem 4.3. (1) If the space (X, τ, I⊛) is ρI⊛-compact then the space (X, τ ⊕ I, I⊛) is I⊛-compact. (2) If the space (X, τ, I⊛) is σI⊛-compact then (X, τ ⊕ I) is compact. (3) If the space (X, τ ⊕ I) is compact then the space (X, τ, I) is ρI-compact. (4) If the space ( X, τ ⊕ I ) is C-compact then the space ( X, τ, I ) is ρC(I)-compact. (5) If ( X, τ ⊕ I ) is QHC then the space ( X, τ, I ) is ρI-QHC. REFERENCES 313 Proof. (1) Suppose that X = ∪ α∈Λ Wα, where Wα ∈ τ ⊕ I for each α ∈ Λ. For all α ∈ Λ, there exist Vα ∈ τ and a collection {Ij}j∈Λα of elements in I, such that Wα = Vα∪ ∪ j∈Λα Ij . Hence X = ∪ α∈Λ Vα ∪ ∪ α∈Λ ∪ j∈Λα Ij . Then X\ ∪ α∈Λ Vα ∈ I⊛ and since (X, τ, I⊛) is ρI⊛-compact, there exists Λ0 ⊆ Λ, finite, with X\ ∪ α∈Λ0 Vα ∈ I⊛. This implies that X\ ∪ α∈Λ0 Wα ∈ I⊛. (3) Suppose that X\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ . There exists I ∈ I such that X\ ∪ α∈Λ Vα = I, and so X = I∪ ∪ α∈Λ Vα. Given that (X, τ ⊕ I) is compact there exists Λ0 ⊆ Λ, finite, with X = I∪ ∪ α∈Λ0 Vα. Hence X\ ∪ α∈Λ0 Vα ⊆ I ∈ I and X\ ∪ α∈Λ0 Vα ∈ I. (4) Suppose that F\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ and F is closed in (X, τ). There exists J ∈ I with F\ ∪ α∈Λ Vα = J , and so F ⊆ J ∪ ∪ α∈Λ Vα. Given that ( X, τ ⊕ I ) is C-compact and F is closed in ( X, τ ⊕ I ) , there exists Λ0 ⊆ Λ, finite, with F ⊆ adhτ⊕I (J) ∪ ∪ α∈Λ0 adhτ⊕I(Vα) ⊆ J ∪ ∪ α∈Λ0 Vα. Hence F\ ∪ α∈Λ0 Vα ⊆ J ∈ I and F\ ∪ α∈Λ0 Vα ∈ I. Parts (2) and (5) have similar demonstrations. □ Acknowledgements The author wishes to express his gratitude to the Escuela Colombiana de Ingeniería Julio Garavito for financing the research that led to this article. Likewise, the author thanks Professor Carlos Abel Alvarez, of the Mathematics Program of this institution, for his support in the final version in LATEX of this paper. References [1] V. Renuka Devi and D. Sivaraj. A generalization of normal spaces. Archivum Math- ematicum, 44:265–270, 2008. [2] S. Jafari and N. Rajesh. Generalized closed sets with respect to an ideal. Eur. Jour. of Pure and App. Math, 4(2):147–151, 2011. REFERENCES 314 [3] D. Jancovic and T. R. Hamlett. New topologies from old via ideals. Amer. Math. Monthly, 97:295–310, 1990. [4] D. Jancovic and T. R. Hamlett. Compatible extensions of ideals. Bollettino U. M. I., (7):453–465, 1992. [5] N. Levine. Generalized closed sets in Topology. Rend. Circ. Mat. Palermo, 19(2):89– 96, 1970. [6] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep. On precontinuous and weak precontinuous mappings. Proc. Math. and Phys. Soc. of Egypt, 53:47–53, 1982. [7] Abd El Monsef, E. F. Lashien, and A. A. Nasef. On I-open sets and I-continuous functions. Kyungpook Math. Jour., 32(1):21–30, 1992. [8] R. L. Newcomb. Topologies which are compact modulo an ideal. PhD thesis, Univ. of Calif. at Santa Barbara. California, 1967. [9] N. R. Pachón. New forms of strong compactness in terms of ideals. Int. Jour. of Pure and App. Math., 106(2):481–493, 2016. [10] N. R. Pachón. ρC(I)-compact and ρI-QHC spaces. Int. Jour. of Pure and App. Math., 108(2):199–214, 2016. [11] J. Porter and J. Thomas. On H-closed and minimal Hausdorff spaces. Trans. Amer. Math. Soc., 138:159–170, 1969. [12] R. Vaidyanathaswamy. The localization theory in set-topology. Proc. Indian Acad. Sci., 20:51–61, 1945. [13] G. Viglino. C-compact spaces. Duke Mathematical Journal, 36(4):761–764, 1969.