COUNTABLE COMPACTNESS MODULO AN IDEAL OF NATURAL NUMBERS

: In this article, we introduce the idea of I -compactness as a covering property through ideals of N and regardless of the I -convergent sequences of points. The frameworks of s -compactness, compactness and sequential compactness are compared to the structure of I -compact space. We began our research by looking at some fundamental characteristics, such as the nature of a subspace of an I -compact space, then investigated its attributes in regular and separable space. Finally, various features resembling ﬁnite intersection property have been investigated, and a connection between I -compactness and sequential I -compactness has been established.


Introduction
The concept of statistical convergence ([17], also [21]) depends on the idea of natural density of subsets of the set of natural number N. The density of a set S ⊆ N is denoted by δ(S) and defined as δ(S) = lim The star operator was introduce by E.K. van Douwen in 1991 [15] as where A is a subset of space X and U is a family of subsets of X.Using star operator the concept of compactness has been generalized in many ways and has been studied by many authors extensively [2][3][4][5][6].In our study we make an attempt to expand this region with the help of Ideal.
In recent days sequential compactness via ideal has been introduced by Singha and Roy [22] under the name of I-compactness, and I-compactness module via an ideal defined on X has also been studied by Gupta and Kaur [18] under the same name.
The purpose of our study is to explore the concept of compactness via ideal of natural number N. We also establish a relation between sequential I-compactness and I-compactness.

Preliminaries
Throughout the paper a space X means a topological space with the corresponding topology τ , '∴' stands for 'therefore' and for other symbols and notions we follow [16].
Definition 1 [16].A topological space X is called a compact space if every open cover of X has a finite subcover, i.e., if for every open cover {U s } s∈S of the space X there exists a finite set Definition 2 [16].A topological space X is called a Lindelöf space if every open cover of X has a countable subcover.
It is known that every compact space is Lindelöf but the converse is not true.
Definition 3 [16].A topological space (X, τ ) is called a countably compact space if every countable open cover of X has a finite sub-cover.
Every compact space is countably compact.But the space W 0 of all countable ordinals is countably compact but not compact [16].The space N of all natural numbers equipped with discrete topology is Lindelöf but it is not countably compact.
Definition 4 [16].A topological space (X, τ ) is said to be sequentially compact space if every sequence in X has a convergent subsequence.
Definition 5 [15].A topological space (X, τ ) will be called a star compact space (in short Stcompact) if for every open cover U of X, there exists a finite subset Definition 6 [8].A topological space (X, τ ) will be called a statistical compact (in short scompact) space if for every countable open cover U = {U n : n ∈ N} of X, there exists a sub-cover

Compactness via ideal
Definition 7. Let I be a non-trivial ideal defined on N. A topological space (X, τ ) will be called an I-compact space if for every countable open cover U = {A n : n ∈ N} of X, there exists a sub-cover Remark 1. Countable compactness is equivalent to I f in -compactness where I f in indicates the ideal of all finite subsets of N. Proposition 1.Every I f in -compact space is a s-compact space.P r o o f.Let (X, τ ) be an I f in -compact space and U = {A n : n ∈ N} be a countable open cover of X. Therefore there exists a sub-cover Example 1. Converse of Proposition 1 may not be true.Indeed there exists a s-compact space which is not

an increasing sequence of open sets by means of inclusion (⊆) and is an open cover of X. It also has a sub-cover
Again suppose that (X, τ ) is I f in -compact and consider the countable open cover Corollary 1.Every Lindelöf I f in -compact space is a compact space.
P r o o f.By Lindelöfness, every open cover has a countable sub-cover.By I f in -compactness, that countable sub-cover will have a finite sub-cover.Hence it will be a compact space.
Theorem 1.Every closed subspace of an I-compact space is an I-compact.P r o o f.Let (A, τ A ) be an arbitrary closed subspace of a I-compact space (X, τ ) and U = {U n : n ∈ N} be a countably infinite cover of (A, τ A ). Then there exists a countable sequence Therefore (A, τ A ) is the I-compact space.
Since f is a continuous surjection mapping and is a countable open cover of X. Again (X, τ ) is I-compact space then there exists a sub-cover, Definition 8. Let (X, τ ) be a topological space and I be a ideal on N. A subset A ⊆ X will be called I-compact subset of X if for every countable cover {U n : n ∈ N} of A by elements of τ there exists a S ∈ I such that A ⊆ n∈S U n .Theorem 3. In a regular space (X, τ ), if A is countable I f in -compact subset of X, then for every closed set B disjoint from A there exists U, V ∈ τ such that A ⊆ U, B ⊆ V and U ∩ V = ∅.P r o o f.Let A = {x n : n ∈ N} be a countable I f in -compact subset of a regular space (X, τ ) and B be an arbitrary closed set disjoint from A, ∴ for every x n ∈ A, x n / ∈ B. But X is a regular space.Therefore there exists It is obvious that which is a contradiction to the fact that Hence the theorem is proven.
P r o o f.In a Hausdörff space, every singleton set {x} is a closed set.So by Theorem 3 the result follows directly.Definition 9. A topological space (X, τ ) will be called sequentially I-compact if every sequence of elements of X has a I-convergent subsequence.Theorem 4. A separable I f in -compact space is a st-compact space P r o o f.Let (X, τ ) be a separable I f in -compact space.Therefore there exists a countable dense subset A = {x n : n ∈ N} of X and U being an arbitrary open cover of X.Using the elements of U we construct a sequence of open sets {U n : n ∈ N} where U n = {U ∈ U : Definition 10.Let I be a ideal on N. A family F = {F n : n ∈ N} of subsets of a space X is said to have I-intersection property if F = ∅ and n∈S F n = ∅ for all S ∈ I.
Theorem 5.For a topological space (X, τ ) and for a non trivial ideal I the following statements are equivalent: (1) For a family G = {G n : n ∈ N} of open sets of X, if for every S ∈ I, G S = {G nα : n α ∈ S} fails to cover X, then G can not cover X. (2) X is an I-compact space.
(3) Every family of countable closed subsets of X with I-intersection property has non-empty intersection.(4) For a family F = {F n : n ∈ N} of closed subsets of X, if F = ∅, then there exists at least one S ∈ I such that n∈S F n = ∅.
(2) ⇔ (3): Let X be an I-compact space, H = {H n : n ∈ N} be a arbitrary family of closed subsets of X having I-intersection property and suppose that H = ∅.
is an countable open cover of X.But X is I-compact.Therefore, there exists a S ∈ I such that n∈S which is a contradiction to the fact that H has I-intersection property, ∴ H = ∅.
Conversely let every countable family of closed subsets with I-intersection property has nonempty intersection. Let Thus the countable family F of closed subsets of X has empty intersection.So it can not have I-intersection property by our assumption.Therefore, there exists a S ∈ I such that n∈S is a sub-cover of U and S ∈ I. Therefore X is an I-compact space.
(3) ⇔ (4): Statement (3) and statement (4) are contrapositive to each other.Therefore, statement (3) and statement (4) are equivalent.Theorem 6. Sequentially I f in -compactness implies I f in -compactness.P r o o f.Let (X, τ ) be an I f in -compact space.Then for every sequence {x n : n ∈ N} of elements of X, there exists is a subsequence {x n k : n ∈ N} which is I-convergent.Let On the other hand let U = {U n : n ∈ N} be a countable open cover of X.So ∃U m ∈ U such that ǫ ∈ U m .Also {n k ∈ N : x n k / ∈ U m } ∈ I f in , suppose {n k 1 , n k 2 , ..., n kp } = {n k ∈ N : x n k / ∈ U m }.But U is a open cover of X. Therefore there exist Now, the collection {U m , U q 1 , U q 2 , ..., U qp } is a sub-cover of U and {m, q 1 , q 2 , ..., q p } is a finite subset of N, i.e. {m, q 1 , q 2 , ..., q p } ∈ I f in , ∴ (X, τ ) is a I f in -compact space.

Conclusion
The paper reveals that I f in -compactness is stronger covering property than statistical compactness, closed subspace of an I-compact space is I-compact, open continuous surjection of an I-compact space is I-compact, a separable I-compact space is a star-compact space.A topological space is I-compact if and only if every family of countable closed subsets of the space which has I-intersection property has non-empty intersection.This study can further be extended for the covering properties like Mengerness and Rothbergerness in the context of modulo an ideal of natural numbers.

Theorem 2 .
Let (X, τ ) be a I-compact space and (Y, σ) be a topological space.If f : (X, τ ) → (Y, σ) is the open continuous surjection mapping, then (Y, σ) is also the I-compact space.P r o o f.Let f : (X, τ ) → (Y, σ) be an open continuous surjection mapping and (X, τ ) is I-compact space.Let {U n : n ∈ N} be a countable open cover of Y .So,