eu NEW FORMS OF STRONG COMPACTNESS IN TERMS OF IDEALS

This work is developed around the concept of compactness modulo an ideal, which was introduced by Newcomb [13], and investigated among others by Hamlett and Jancovic [6], Rancin [15] Gupta and Kaur [5]. Some classes of strong compactness modulo an ideal were studied by Newcomb [13], Hamlett et al. [7], Abad El Monsef [2], Nasef and Noiri [11], Nasef [12] and Hosny [8]. Also, some classes of weak compactness, module an ideal, were studied by Gupta and Noiri [4]. The purpose of this paper is to introduce and investigate the classes of ρIcompact and σI-compact spaces. The behavior of this spaces under certain kinds of functions also is investigated. An ideal on a set X is a nonempty family I of subsets of X such that:

The purpose of this paper is to introduce and investigate the classes of ρIcompact and σI-compact spaces.The behavior of this spaces under certain kinds of functions also is investigated.
An ideal on a set X is a nonempty family I of subsets of X such that: (i) If B ∈ I and A ⊆ B ⊆ X, then A ∈ I.
(ii) If A ∈ I and B ∈ I, then A ∪ B ∈ I.
For example, if X is a set and B ⊆ X, then the following sets are ideals on X: If (X, τ ) is a topological space and I is an ideal on X, then (X, τ, I) is called an ideal space.
A subset A of a space (X, τ ) is said to be g-closed [10] if A ⊆ U whenever A ⊆ U and U ∈ τ .It is clear that every closed set is g-closed, but the converse is not true.
If (X, τ, I) is an ideal space, (Y, β) is a topological space and f : If (X, τ, I) is an ideal space, the set B = {U \I : U ∈ τ and I ∈ I} is a base for a topology τ * , finer than τ .
If (X, τ ) is a topological space and A ⊆ X then A and int (A) will, respectively, denote the closure and interior of A in (X, τ ).

ρI-compact spaces
We recall that a subset A of an ideal space (X, τ, I) is said to be I-compact [13], if for every open cover {V α } α∈Λ of A by elements of τ , there exists Λ 0 ⊆ Λ, finite, such that A\ It is clear that (X, τ ) is compact if and only if (X, τ, {∅}) is ρ {∅}-compact, and that if (X, τ, I) is ρI-compact then (X, τ, I) is I-compact.The converse is not true.
α∈Λ is an open cover of X, then there exists α 0 ∈ Λ with V α 0 = X, and so X\V α 0 ∈ I.
ii) (X, τ, I) is not ρI-compact, because X\ r>0 (r, +∞) = {0} ∈ I, but if n is a positive integer and 0 < r In the example 3.1 we show a ρI-compact ideal space.
Definition 2.2 A subset A of an ideal space (X, τ, I) is said to be Ig-closed and A\U ∈ I, we have that A ⊆ U , and so U = X and A ⊆ U .
Next we study the behavior of some types of subspaces of a ρI-compact space.V α ∈ I.This implies that A\ Theorem 2.5 Let (X, τ, I) be an ideal space and A ⊆ X. Suppose that for all U ∈ τ , if A\U ∈ I then there exists B ⊆ X such that B is ρI-compact, A ⊆ B and B\U ∈ I. Then A is ρI-compact.
Proof.(→) Let {V α } α∈Λ be a family of open subsets of X such that B\ α∈Λ V α ∈ I. Then A\ α∈Λ V α ∈ I, and given that A is ρI-compact there exists Λ 0 ⊆ Λ, finite, such that A\ and so A\ The following theorem is consequence of [Theorem 2.2, [5]].
Theorem 2.7 Let (X, τ, I) be an ideal space such that (X, Theorem 2.8 Let (X, τ, I) be a ρI-compact space such that (X, τ ) is T 2 .If F and G are disjoint Ig-closed subsets of X, then there exist disjoint open subsets U and V of X, such that F \ U ∈ I and G \ V ∈ I.

Proof. The result is clear if
Theorem 2.3 implies that F and G are ρI-compact subsets of X.
We choose g ∈ G, arbitrary but fixed.For all f ∈ F there exist disjoint Now, since that G\ g∈G W g = ∅ ∈ I and G is ρI-compact, there exists and Now we study the behavior of ρI−compactness under certain types of functions.
If f : X → Y is an inyective function and J is an ideal on Y , then the set f −1 (J ) = f −1 (J) : J ∈ J is an ideal on X [13].

σI-compact spaces
In this section we present a strong form of ρI-compactness.Some properties of these spaces are also presented.
Example 3.1 (1) Let X = Z + , τ = {A ⊆ X : X\A is finite} ∪ {∅} and I = I f .Then: (a) The ideal space (X, τ, I) is ρI-compact, because if {F i } i∈Λ is a family of closed subsets of X with i∈Λ F i ∈ I, then there exists i 0 ∈ Λ such that F i 0 = X.F n k = ∅.
In the example 4.1 we show a σI-compact ideal space.

Theorem 2 . 4
If A and B are ρI-compact subsets of an ideal space (X, τ, I), then A ∪ B is ρI-compact.Proof.Let {V α } α∈Λ be a family of open subsets such that (A ∪ B)