C-COMPACTNESS MODULO AN IDEAL

In the present paper, we consider a topological space equipped with an ideal, a theme that has been treated by Vaidyanathaswamy [15] and Kuratowski [6] in their classical texts. An ideal on a set X is a nonempty subset of P(X), the power set of X , which is closed for subsets and finite unions. An ideal is also called a dual filter. {φ} and P(X) are trivial examples of ideals. Some useful ideals are (i) f , the ideal of all finite subsets of X , (ii) c, the ideal of all countable subsets of X , (iii) n , the ideal of all nowhere dense subsets in a topological space (X ,τ), and (iv) s, the set of all scattered sets in (X ,τ). For an ideal on X and A⊂ X , we denote the ideal {I ∩A : I ∈ } by A. A topological space (X ,τ) with an ideal on X is denoted by (X ,τ, ). For a subset A⊆ X , A∗( ,τ) (called the adherence of A modulo an ideal ) or A∗( ) or just A∗ is the set {x ∈ X : A∩U / ∈ for every open neighborhood U of x}. A∗( ,τ) has been called the local function of A with respect to in [6]. It is easy to see that (i) for the ideal {φ}, A∗ is the closure of A, (ii) for the ideal P(X), A∗ is φ, and (iii) for ideal f , A∗ is the set of all ω-accumulation points of A. For general properties of the operator ∗, we refer the readers to [5, 14]. Observe that the operator cl∗ : P(X)→P(X) defined by cl∗(A)=A∪A∗ is a Kuratowski closure operator on X and hence generates a topology τ∗( ) or just τ∗ on X finer than τ. As has already been observed, τ∗({φ}) = τ and τ∗(P(X)) = the discrete topology. A description of open sets in τ∗( ) as given in Vaidyanathaswamy [15] is given in the following.


Introduction
In the present paper, we consider a topological space equipped with an ideal, a theme that has been treated by Vaidyanathaswamy [15] and Kuratowski [6] in their classical texts.An ideal Ᏽ on a set X is a nonempty subset of P(X), the power set of X, which is closed for subsets and finite unions.An ideal is also called a dual filter.{φ} and P(X) are trivial examples of ideals.Some useful ideals are (i) Ᏽ f , the ideal of all finite subsets of X, (ii) Ᏽ c , the ideal of all countable subsets of X, (iii) Ᏽ n , the ideal of all nowhere dense subsets in a topological space (X,τ), and (iv) Ᏽ s , the set of all scattered sets in (X,τ).For an ideal Ᏽ on X and A ⊂ X, we denote the ideal {I ∩ A : I ∈ Ᏽ} by Ᏽ A .
A topological space (X,τ) with an ideal Ᏽ on X is denoted by (X,τ,Ᏽ).For a subset A ⊆ X, A * (Ᏽ,τ) (called the adherence of A modulo an ideal Ᏽ) or A * (Ᏽ) or just A * is the set {x ∈ X : A ∩ U / ∈ Ᏽ for every open neighborhood U of x}.A * (Ᏽ,τ) has been called the local function of A with respect to Ᏽ in [6].It is easy to see that (i) for the ideal {φ}, A * is the closure of A, (ii) for the ideal P(X), A * is φ, and (iii) for ideal Ᏽ f , A * is the set of all ω-accumulation points of A. For general properties of the operator * , we refer the readers to [5,14].
Observe that the operator cl * : P(X)→P(X) defined by cl * (A)=A ∪ A * is a Kuratowski closure operator on X and hence generates a topology τ * (Ᏽ) or just τ * on X finer than τ.As has already been observed, τ * ({φ}) = τ and τ * (P(X)) = the discrete topology.A description of open sets in τ * (Ᏽ) as given in Vaidyanathaswamy [15] is given in the following.
Theorem 1.1.If τ is a topology and Ᏽ is an ideal, both defined on X, then β = β(τ,Ᏽ) = {V − I : V ∈ τ, I ∈ Ᏽ} is a base for the topology τ * (Ᏽ) on X. (1.1) Ideals have been used frequently in the fields closely related to topology, such as real analysis, measure theory, and lattice theory.Some interesting illustrations of τ * (Ᏽ) are as follows [5].
(1) If τ is the topology generated by the partition {{2n − 1,2n} : n ∈ N} on the set N of natural numbers, then τ * (Ᏽ f ) is the discrete topology.(2) If τ is the indiscrete topology on a set X, then τ * (Ᏽ f ) is the cofinite topology on X, and τ * (Ᏽ c ) is the co-countable topology on X.If for a fixed point p ∈ X, Ᏽ denotes the ideal {A ⊂ X : p / ∈ A}, then τ * (Ᏽ) is the particular point topology on X.
(3) For any topological space (X,τ), τ * (Ᏽ n ) is the τ α topology of Njȧstad [10].(4) If τ is the usual topology on the real line R and Ᏽ is the ideal of all subsets of Lebesgue measure zero, then τ * -Borel sets are precisely the Lebesgue measurable sets of R.

Quasi-H-closed modulo an ideal space
The concept of compactness modulo an ideal was introduced by Newcomb [9] and has been studied among others by Rancin [11], and Hamlett and Janković [3].A space (X,τ) is defined to be compact modulo an ideal Ᏽ on X or just (Ᏽ) compact space if for every open cover ᐁ of X, there is a finite subfamily {U 1 ,U 2 ,...,U n } such that X − n i=1 U i ∈ Ᏽ.In this section, we define quasi-H-closedness modulo an ideal and study some of its properties.In the process, we get some interesting characterizations of quasi-H-closed spaces.
Definition 2.1.Let (X,τ) be a topological space and Ᏽ an ideal on X. X is quasi-H-closed modulo Ᏽ or just (Ᏽ) QHC if for every open cover ᐁ of X, there is a finite subfamily Such a subfamily is said to be proximate subcover modulo Ᏽ or just (Ᏽ) proximate subcover.
A subset A of a topological space (X,τ) is said to be preopen [8] if A ⊂ int(cl(A)).The collection of all preopen sets of a space (X,τ) is denoted by PO(X).An ideal Ᏽ of subsets of a topological space (X,τ) is said to be codense [1] if the complement of each of its members is dense.Note that an ideal Ᏽ is codense if and only if Ᏽ ∩ τ = {φ}.Codense ideals are called τ-boundary ideals in [9].An ideal Ᏽ of subsets of a topological space (X,τ) is said to be completely codense [1] if Ᏽ ∩ PO(X) = {φ}.Obviously, every completely codense ideal is codense.Note that if (R,τ) is the set R of real numbers equipped with the usual topology τ, then Ᏽ c is codense but not completely codense ideal.It is proved in [1] that an ideal Ᏽ is completely codense if and only if Ᏽ ⊂ Ᏽ n .
From the discussion of Section 1, the proof of the following theorem is immediate.Theorem 2.2.For a space (X,τ), the following are equivalent: The significance of condition in (e) may be seen by considering the set R of real numbers equipped with the usual topology τ.If A is a finite subset of R and Ᏽ is the ideal of all subsets of R − A, then (R,τ) is (Ᏽ)QHC, but not quasi-H-closed.
A family Ᏺ of subsets of X is said to have the finite-intersection property modulo an ideal Ᏽ on X or just (Ᏽ)FIP if the intersection of no finite subfamily of Ᏺ is a member of Ᏽ .Recall that a subset in a space is called regular open if it is the interior of its own closure.The complement of a regular open set is called regular closed.It is proved in [12] that for completely codense ideal Ᏽ on a space (X,τ), the collections of regular open sets of (X,τ) and (X,τ * ) are same.The following theorem contains a number of characterizations of (Ᏽ)QHC spaces.Since the proof is similar to that of a theorem in the next section, we omit it.
Theorem 2.3.For a space (X,τ) and an ideal Ᏽ on X, the following are equivalent: It follows from a result in [13] that τ and τ * (Ᏽ) have the same regular open sets, where Ᏽ is a completely codense ideal on (X,τ).In particular, if U ∈ τ * , then cl(U) = cl * (U).Using this observation along with the previous theorem, we have the following.
Combining this result with Theorem 2.2, we have the following.
Corollary 2.5.Let (X,τ) be a space and Ᏽ a completely codense ideal on X.Then the following are equivalent: The last equivalence follows because τ α = τ * (Ᏽ n ), where Ᏽ n is the ideal of nowhere dense sets in X.

C-compact modulo an ideal space
In this section, we generalize the concepts of C-compactness of Viglino [16] and compactness modulo an ideal due to Newcomb [9] and Rancin [11].A space (X,τ) is said to be C-compact if for each closed set A and each τ-open covering ᐁ of A, there exists a finite subfamily {U 1 ,U 2 ,U 3 ,...,U n } such that A ⊂ n i=1 cl(U i ).Definition 3.1.Let (X,τ) be a topological space and Ᏽ an ideal on X. (X,τ) is said to be Ccompact modulo Ᏽ or just C(Ᏽ)-compact if for every closed set A and every τ-open cover ᐁ of A, there is a finite subcollection Also from the definition in Section 1, we have the following.
Theorem 3.2.For a space (X,τ), the following are equivalent: For n and m in the set N of positive integers, let Y denote the subset of the plane consisting of all points of the form (1/n,1/m) and the points of the form (1/n,0). Let X = Y ∪ {∞}.Topologize X as follows: let each point of the form (1/n,1/m) be open.
Partition N into infinitely many infinite-equivalence classes, {Z i } ∞ i=1 .Let a neighborhood system for the point (1/i,0) be composed of all sets of the form G ∪ F, where for some k ∈ N. Let a neighborhood system for the point ∞ be composed of sets of the form X\T, where for some k ∈ N. It is shown in [16] that X is a C-compact space which is not compact.In view of Theorem 3.2, such a space is C(Ᏽ f )-compact, but not (Ᏽ f ) compact.
Example 3.4.Let X = R + ∪ {a} ∪ {b}, where R + denotes the set of nonnegative real numbers and a, b are two distinct points not in R + .Let W(a) = {V ⊂ X : V = {a} ∪ ∞ r=m (2r, 2r + 1)}, where m is a nonnegative integer, be a neighborhood system for the point a.
where m is a nonnegative integer, be a neighborhood system for the point b.Let R + , with the usual topology, be imbedded in X. Viglino [16] has shown that the space X is not C-compact.If A is a finite subset of X, then (X,τ) is C(Ᏽ)-compact, where Ᏽ is the ideal of all subsets of X − A.
In view of Examples 3.3 and 3.4, it is clear that the implications shown after Definition 3.1 are, in general, irreversible.
It is proved in [3] that if (X,τ) is quasi-H-closed and Ᏽ is an ideal such that In view of this discussion, we have the following.Theorem 3.5.For a space (X,τ), the following are equivalent: A space (X,τ) is said to be Baire if the intersection of every countable family of open sets in (X,τ) is dense.It is noted in [5] that a space (X,τ) is Baire if and only if τ ∩ Ᏽ m = {φ}, where Ᏽ m is the ideal of meager (first category) subsets of (X,τ).Thus, in view of the above theorem, a Baire space (X,τ) is C(Ᏽ m )-compact if and only if it is quasi-H-closed.
We now give some characterizations of C(Ᏽ)-compact spaces.
Theorem 3.6.Let (X,τ) be a space and let Ᏽ be an ideal on X.Then the following are equivalent: This set in Ᏽ is easily seen to be n i=1 {int(F i ) ∩ A}. (b)⇒(c).This is easy to be established.(c)⇒(a).Let A be a closed subset, let ᐁ be an open cover of A with the property that for no finite subfamily Let {U 1 ,U 2 ,U 3 ,...,U n } be a finite subcollection of ᐁ such that (X − V ) − n i=1 cl(U i ) ∈ Ᏽ.However, the last set is X − (V ∪ { n i=1 cl(U i )}).(g)⇒(a).Let A be a closed subset of X and ᐁ an open covering of A. If H denotes the union of members of ᐁ, then F = X − H is a closed set and X − A is an open neighborhood of F. Also ᐁ is an open cover of X − F. By hypothesis, there is a finite subcollection However, this set in Ᏽ is nothing but A − n i=1 cl(U i ).(a)⇒(h).Suppose A is a closed set and Ꮾ is any open filter base on X with {B ∩ A : is an open cover of A. By the hypothesis, there exists a finite subfamily {X − cl(B i ) : i = 1,2,3,...,n} such that (h)⇒(a).Suppose that (X,τ) is not C(Ᏽ)-compact.Then there exist a closed subset A of X and an open cover ᐁ of A such that for any finite subfamily {U 1 ,U 2 ,U 3 ,...,U n } of ᐁ, we have A − n i=1 cl(U i ) / ∈ Ᏽ.We may assume that ᐁ is closed under finite unions.Then the family Ꮾ = {X − cl(U) : U ∈ ᐁ} is an open filter base on X such that {B ∩ A : B ∈ Ꮾ} ⊂ P(A) − Ᏽ.So, by the hypothesis, {cl(X − cl(U)) : U ∈ ᐁ} ∩ A = φ.Let x be a point in the intersection.Then Next we characterize C(Ᏽ)-compact spaces using some weaker forms of filter base convergence.
Definition 3.7.A filter base Ꮾ is said to be (Ᏽ) adherent convergent if for every neighborhood G of the adherent set of Ꮾ, there exists an element Clearly, every adherent convergent filter base is (Ᏽ) adherent convergent and a filter base is adherent convergent if and only if it is ({φ}) adherent convergent.

Theorem 3.8. A space (X,τ) is C(Ᏽ)-compact if and only if every open filter base on P(X) −
Ᏽ is (Ᏽ) adherent convergent.
Proof.Let (X,τ) be C(Ᏽ)-compact and let Ꮾ be an open filter base on P(X) − Ᏽ with A as its adherent set.Let G be an open neighborhood of A.
and so by the hypothesis, it admits a finite subfamily Since Ꮾ is a filter base and ᐁ is an open cover of A and for U ∈ ᐁ, U ⊂ int(cl(U)).Therefore, the adherent set of Ꮾ is contained in X − A, which is an open set.By the hypothesis, there exists an element This however contradicts our assumption.This completes the proof.
Herrington and Long [4] characterized C-compact spaces using r-convergence of filters and nets.We obtain similar results for C(Ᏽ)-compact spaces in the next definition.Definition 3.9.Let X be a space, φ = A ⊂ X, and let Ꮾ be a filter base on Similarly, a net ϕ : It is known [4] that convergence (accumulation) for filter bases and nets implies rconvergence (r-accumulation), but the converse is not true.Theorem 3.10.For a space (X,τ) and an ideal Ᏽ on X, the following are equivalent: (b)⇔(c).This follows in view of parts (a), ( ⇒(a).If possible, let X be not C(Ᏽ)-compact.Then by Theorem 3.6(f), there exist a closed set A and a collection Ᏺ of regular closed sets with the property that for every finite subcollection A for all possible finite subfamilies {F 1 ,F 2 ,F 3 ,...,F n } of Ᏺ forms a filter base on P(A) − Ᏽ.By (b), this filter base raccumulates to some a ∈ A, that is, for each open set U(a) containing a and for each F ∈ Ᏺ, cl(U(a)) ∩ (int(F) ∩ A) = φ.However, a ∈ A and A ∩ {F : F ∈ Ᏺ} = φ imply that there is some F = F(a) ∈ Ᏺ such that a / ∈ F(a).Then X − F(a) is an open set containing a such that cl(X − F(a)) ∩ (int(F(a)) ∩ A) = φ.This is a contradiction.
(b)⇔(d).This follows using standard arguments about nets and filters.
If in the above theorem, A is replaced by the whole space X, we get the characterizations of (Ᏽ) QHC spaces.If in addition we consider completely codense ideal Ᏽ, we get the characterizations of quasi-H-closed spaces.

C(Ᏽ)-compact spaces and functions
A function f : (X,τ) − (Y ,σ) is said to be θ-continuous [2] at a point x ∈ X if for every open set V of Y containing f (x), there exists an open set U of X containing x such that f (cl(U)) ⊆ cl(V ).A function f : (X,τ) − (Y ,σ) is said to be θ-continuous if f is θcontinuous for every x ∈ X.The concept of θ-continuity is weaker than that of continuity.An important property of C-compact spaces is that a continuous function from a C-compact space to a Hausdorff space is closed.We prove the following more general results.Proof.Let A be any closed set in X and a / ∈ f (A).For each x ∈ A, there exists a σ-open set V y containing y = f (x) such that a / ∈ cl(V y ).Now because f is θ-continuous, there exists an open set U x containing x such that f (cl(U x )) ⊆ cl(V y ).The family {U x : x ∈ A} is an open cover of A. Therefore, there exists a finite subfamily {U xi : i = 1,2,...,n} such that Proof.Let A be any closed subset of (Y ,σ) and Hence, by the hypothesis, there exists a finite subcollection Proof.This follows from Theorem 4.3.

C(Ᏽ)-compact spaces and subspaces
In this section, we introduce three types of C(Ᏽ)-compact subsets and use them to obtain new characterizations of C(Ᏽ)-compact spaces and a characterization of maximal C(Ᏽ)compact spaces.
Definition 5.1.Let (X,τ) be a space and Ᏽ an ideal on Some useful results about such subspaces are contained in the following theorem.The proofs are easy to establish.Theorem 5.2.Let (X,τ) be a space and Ᏽ an ideal on X.Then Finally, we obtain a characterization of a maximal C(Ᏽ)-compact space.Recall that a space (X,τ) with property P is said to be maximal P if there is no topology σ on X which has property P and is strictly finer than τ.For a topological space (X,τ) and a subset A of X, τ(A) = {U ∪ (V ∩ A) : U,V ∈ τ} is a topology called simple extension [7] of τ by A. τ(A) is strictly finer than τ if and only if A / ∈ τ.
Theorem 5.9.A topological space (X,τ) is maximal C(Ᏽ)-compact if and only if for every subset A of X such that A is closure C(Ᏽ)-compact and X − A is C(Ᏽ)-compact relative to τ, one has A ∈ τ.
Conversely, let (X,τ) be not maximal C(Ᏽ)-compact.Then there is a C(Ᏽ)-compact topology σ on X which is strictly finer than τ.Let A ∈ σ − τ.Then A is σ-closure C(Ᏽ)compact by Theorem 5.8.Since the property of closure C(Ᏽ)-compact is carried over to coarser topologies, A is τ-closure C(Ᏽ)-compact.Also X − A is C(Ᏽ)-compact relative to σ and hence C(Ᏽ)-compact relative to τ.By the hypothesis, then A ∈ τ, a contradiction.
Remark 5.10.The readers can generalize the above concepts in bitopological spaces to unify various types of compactness.
(a)⇒(d).This is obvious.The proofs for (d)⇒(e)⇒(f)⇒(d) are parallel to (a)⇒(b)⇒(c)⇒(a), respectively.(a)⇒(g).Let A be a closed set, V an open neighborhood of A, and ᐁ an open cover of X − A. Since X − V ⊂ X − A, ᐁ is also an open cover of the closed set X − V .
let (X,τ) be not C(Ᏽ)-compact, and let A be a closed set, and ᐁ an open cover of A such that for no finite subfamily {U 1 ,U 2 ,U 3 ,...,U n } of ᐁ, one has A − n i=1 cl(U i ) ∈ Ᏽ.Without loss of generality, we may assume that ᐁ is closed for finite unions.Therefore, Ꮾ = {X − cl(U) : U ∈ ᐁ} becomes an open filter base on P (a) (X,τ) is C(Ᏽ)-compact; (b) for each closed set A, each filter base Ꮾ on P(A) − Ᏽ r-accumulates to some a ∈ A; (c) for each closed set A, each maximal filter base ᏹ on P(A) − Ᏽ r-converges to some a ∈ A; (d) for each closed set A, each net ϕ on P(A) − Ᏽ r-accumulates to some a ∈ A. Proof.(a)⇒(b).Suppose there exist a closed set A and a filter base Ꮾ on P(A) − Ᏽ which does not r-accumulate to any a ∈ A. Then for each a ∈ A, there exists an open set U(a) containing a and a B(a) ∈ Ꮾ such that B(a) ∩ cl(U(a)) = φ.Then {U(a) : a ∈ A} is an open cover of the closed set A. By (a), there exists a finite subcollection {U(a i )
1) be a τ(A)-open cover of K. Then ν = {U α : α ∈ Δ} is a τ-open cover of K ∩ (X − A) = (K 1 ∪ K 2 ) ∩ (X − A).Since, by assumption, X − A is C(Ᏽ)-compact relative to τ, we have a finite subcollection {U α1 ,U α2 ,U α3 ,...,U αn } of ν such that K ∩ (X − A) − n i=1 cl(U αi ) ∈ Ᏽ.Since τ(A) is finer than τ,this subcollection is τ(A)-open and K for each family Ᏺ of closed sets having empty intersection, there is a finite subfamily {F 1 ,F 2 ,F 3 ,...,F n } such that n i=1 int(F i ) ∈ Ᏽ; (c) for each family Ᏺ of closed sets such that {int(F) : F ∈ Ᏺ} has (Ᏽ)FIP, one has ∩{F : ∈ Ᏺ} = φ, there exists a finite subfamily {F 1 ,F 2 ,F 3 ,...,F n } such that (int(F i )) ∩ A ∈ Ᏽ; (c)for each closed set A and each family Ᏺ of closed sets such that {int(F) ∩ A : F ∈ Ᏺ} has (Ᏽ)FIP, one has ∩{F ∩ A : F ∈ Ᏺ} = φ; (d) for each closed set A and each regular open cover ᐁ of A, there exists a finite subcollection {U 1 ,U 2 ,U 3 ,...,U n } such that A − n i=1 cl(U i ) ∈ Ᏽ; (e) for each closed set A and each family Ᏺ of regular closed sets such that {F ∩ A : F ∈ Ᏺ} = φ, there is a finite subfamily {F 1 ,F 2 ,F 3 ,...,F n } such that n i=1 (int(F i )) ∩ A ∈ Ᏽ; (f) for each closed set A and each family Ᏺ of regular closed sets such that {int(F) ∩ A : F (h) for each closed set A and each open filter base Ꮾ on X such that {B ∩ A : B ∈ Ꮾ} ⊂ P(X) − Ᏽ, one has {cl(B) : B ∈ Ꮾ} ∩ A = φ.Proof.(a)⇒(b).Let (X,τ) be C(Ᏽ)-compact, A a closed subset, and Ᏺ a family of closed subsets with ∩{F ∩ A : F ∈ Ᏺ} = φ.Then {X − F : F ∈ Ᏺ} is an open cover of A and hence admits a finite subfamily {X − F i and each family Ᏺ of closed subsets of X such that {F A : F ∈ Ᏺ} has (Ᏽ)FIP, one has {F ∩ A : F ∈ Ᏺ} = φ; (g) for each closed set A, each open cover ᐁ of X − A and each open neighborhood V of A, there exists a finite subfamily {U 1 ,U 2 ,U 3 ,...,U n } of ᐁ such that X − (V ∪ ( n i=1 cl(U i ))) ∈ Ᏽ;