C
-COMPACTNESS MODULO AN IDEAL
M. K. GUPTA AND T. NOIRI
Received 24 January 2006; Revised 30 March 2006; Accepted 4 April 2006
We investigate the concepts of quasi-H-closed modulo an ideal which generalizes quasi-H-closedness andC-compactness modulo an ideal which simultaneously generalizes C-compactness and C-compactness modulo an ideal. We obtain a characterization of maxi-malC-compactness modulo an ideal. Preservation ofC-compactness modulo an ideal by functions is also investigated.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. 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 idealᏵon a setXis a nonempty subset ofP(X), the power set ofX, which is closed for subsets and finite unions. An ideal is also called a dual filter.{φ}andP(X) are trivial examples of ideals. Some useful ideals are (i)Ᏽf, the ideal of all finite subsets ofX, (ii) Ᏽc, the ideal of all countable subsets ofX, (iii)Ᏽn, the ideal of all nowhere dense subsets in a topological space (X,τ), and (iv)Ᏽs, the set of all scattered sets in (X,τ). For an ideal
ᏵonXandA⊂X, we denote the ideal{I∩A:I∈Ᏽ}byᏵA.
A topological space (X,τ) with an idealᏵonX is denoted by (X,τ,Ᏽ). For a subset A⊆X,A∗(Ᏽ,τ) (called the adherence of A modulo an idealᏵ) orA∗(Ᏽ) or justA∗is the set{x∈X:A∩U /∈Ᏽfor every open neighborhoodUofx}.A∗(Ᏽ,τ) has been called the local function ofAwith respect toᏵin [6]. It is easy to see that (i) for the ideal{φ}, A∗is the closure ofA, (ii) for the idealP(X),A∗isφ, and (iii) for idealᏵ
f,A∗is the set
of allω-accumulation points ofA. 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 onXand hence generates a topologyτ∗(Ᏽ) or justτ∗onXfiner than τ. As has already been observed,τ∗({φ})=τandτ∗(P(X))=the discrete topology. A description of open sets in τ∗(Ᏽ) as given in Vaidyanathaswamy [15] is given in the following.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 78135, Pages1–12
Theorem 1.1. Ifτis a topology andᏵis an ideal, both defined onX, then
β=β(τ,Ᏽ)= {V−I:V∈τ,I∈Ᏽ}is a base for the topologyτ∗(Ᏽ) onX. (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 ofτ∗(Ᏽ) are as follows [5].
(1) Ifτis the topology generated by the partition{{2n−1, 2n}:n∈N}on the setN
of natural numbers, thenτ∗(Ᏽf) is the discrete topology.
(2) Ifτis the indiscrete topology on a setX, thenτ∗(Ᏽf) is the cofinite topology on
X, andτ∗(Ᏽc) is the co-countable topology onX. If for a fixed pointp∈X,Ᏽ denotes the ideal{A⊂X:p /∈A}, thenτ∗(Ᏽ) is the particular point topology onX.
(3) For any topological space (X,τ),τ∗(Ᏽn) is theταtopology of Nj˙astad [10].
(4) Ifτ is the usual topology on the real lineRandᏵis the ideal of all subsets of Lebesgue measure zero, thenτ∗-Borel sets are precisely the Lebesgue measurable sets ofR.
2. 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 Jankovi´c [3]. A space (X,τ) is defined to be compact modulo an idealᏵonX or just (Ᏽ) compact space if for every open coverᐁofX, there is a finite subfamily{U1,U2,...,Un}such thatX−ni=1Ui∈
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 andᏵan ideal onX.Xis quasi-H-closed modulo Ᏽor just (Ᏽ) QHC if for every open coverᐁ ofX, there is a finite subfamily {U1,U2,...,Un}ofᐁsuch thatX−
n
i=1cl(Ui)∈Ᏽ. Such a subfamily is said to be
proxi-mate subcover moduloᏵor just (Ᏽ) proximate subcover.
A subsetAof a topological space (X,τ) is said to be preopen [8] ifA⊂int(cl(A)). The collection of all preopen sets of a space (X,τ) is denoted by PO(X). An idealᏵof 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 idealᏵis codense if and only ifᏵ∩τ= {φ}. Codense ideals are calledτ-boundary ideals in [9]. An idealᏵ of subsets of a topological space (X,τ) is said to be completely codense [1] ifᏵ∩PO(X)= {φ}. Obviously, every completely codense ideal is codense. Note that if (R,τ) is the setRof real numbers equipped with the usual topologyτ, thenᏵcis codense but not completely codense ideal. It is proved in [1] that an idealᏵis completely codense if and only ifᏵ⊂Ᏽn.
From the discussion ofSection 1, the proof of the following theorem is immediate. Theorem 2.2. For a space (X,τ), the following are equivalent:
(b) (X,τ) is ({φ}) QHC; (c) (X,τ) is (Ᏽf) QHC; (d) (X,τ) is (Ᏽn) QHC;
(e) (X,τ) is (Ᏽ) QHC for every codense idealᏵ.
The significance of condition in (e) may be seen by considering the setRof real num-bers equipped with the usual topologyτ. IfAis a finite subset ofRandᏵis the ideal of all subsets ofR−A, then (R,τ) is (Ᏽ) QHC, but not quasi-H-closed.
A familyᏲof subsets ofXis said to have the finite-intersection property modulo an ideal
ᏵonXor just (Ᏽ) FIP if the intersection of no finite subfamily ofᏲis a member ofᏵ. 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 idealᏵon 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 idealᏵonX, the following are equivalent: (a) (X,τ) is (Ᏽ) QHC;
(b) for each familyᏲof closed sets having empty intersection, there is a finite subfamily {F1,F2,F3,...,Fn}such thatni=1int(Fi)∈Ᏽ;
(c) for each familyᏲof closed sets such that{int(F) :F∈Ᏺ}has (Ᏽ) FIP, one has∩{F: F∈Ᏺ} =φ;
(d) every regular open cover has a finite (Ᏽ) proximate subcover;
(e) for each familyᏲof nonempty regular closed sets having empty intersection, there is a finite subfamily{F1,F2,F3,...,Fn}such that
n
i=1int(Fi)∈Ᏽ;
(f) for each collectionᏲof nonempty regular closed sets such that{int(F) :F∈Ᏺ}has (Ᏽ) FIP, one has{F:F∈Ᏺ} =φ;
(g) for each open filter baseᏮon P (X)−Ᏽ,{cl(B) :B∈Ꮾ} =φ; (h) every open ultrafilter onP(X)−Ᏽconverges.
It follows from a result in [13] thatτandτ∗(Ᏽ) have the same regular open sets, where
Ᏽis a completely codense ideal on (X,τ). In particular, ifU∈τ∗, then cl(U)=cl∗(U). Using this observation along with the previous theorem, we have the following.
Theorem 2.4. LetᏵbe a completely codense ideal on a space (X,τ). Then (X,τ) is (Ᏽ) QHC if and only if (X,τ∗) is (Ᏽ) QHC.
Combining this result withTheorem 2.2, we have the following.
Corollary 2.5. Let (X,τ) be a space and Ᏽa completely codense ideal onX. Then the following are equivalent:
(a) (X,τ) is quasi-H-closed; (b) (X,τ∗) is quasi-H-closed; (c) (X,τα) is quasi-H-closed.
3.C-compact modulo an ideal space
In this section, we generalize the concepts ofC-compactness of Viglino [16] and com-pactness modulo an ideal due to Newcomb [9] and Rancin [11]. A space (X,τ) is said to beC-compact if for each closed setAand eachτ-open coveringᐁofA, there exists a finite subfamily{U1,U2,U3,...,Un}such thatA⊂
n
i=1cl(Ui).
Definition 3.1. Let (X,τ) be a topological space andᏵan ideal onX. (X,τ) is said to beC -compact moduloᏵor justC(Ᏽ)-compact if for every closed setAand everyτ-open cover
ᐁofA, there is a finite subcollection{U1,U2,U3,...,Un}such thatA−
n
i=1cl(Ui)∈Ᏽ.
It follows from the definition that
compact (Ᏽ) compact
C-compact C(Ᏽ)-compact
quasi-H-closed (Ᏽ) QHC
(3.1)
Also from the definition inSection 1, we have the following. Theorem 3.2. For a space (X,τ), the following are equivalent:
(a) (X,τ) isC-compact; (b) (X,τ) isC({φ})-compact; (c) (X,τ) isC(Ᏽf)-compact.
Example 3.3. Fornandmin the setNof positive integers, letYdenote 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). LetX=Y∪ {∞}. TopologizeXas follows: let each point of the form (1/n, 1/m) be open. PartitionNinto infinitely many infinite-equivalence classes,{Zi}∞i=1. Let a neighborhood
system for the point (1/i, 0) be composed of all sets of the formG∪F, where
G=1 i, 0
∪1
i, 1 m
:m≥k
,
F=1 n,
1 m
:m∈Zi,n≥k
(3.2)
for somek∈N. Let a neighborhood system for the point∞be composed of sets of the formX\T, where
T=
1 n, 0
:n∈N∪
k
i=1
1 i, 1 m
:m∈N∪ 1 n, 1 m
:m∈Zi,n∈N
(3.3)
Example 3.4. LetX=R+∪ {a} ∪ {b}, whereR+denotes the set of nonnegative real
num-bers anda,bare two distinct points not inR+. LetW(a)= {V⊂X:V= {a} ∪∞
r=m(2r,
2r+ 1)}, wheremis a nonnegative integer, be a neighborhood system for the pointa. LetW(b)= {V⊂X:V= {b} ∪∞r=m(2r−1, 2r)}, wheremis a nonnegative integer, be a neighborhood system for the pointb. LetR+, with the usual topology, be imbedded in
X. Viglino [16] has shown that the spaceXis notC-compact. IfAis a finite subset ofX, then (X,τ) isC(Ᏽ)-compact, whereᏵis the ideal of all subsets ofX−A.
In view of Examples3.3and3.4, it is clear that the implications shown afterDefinition 3.1are, in general, irreversible.
It is proved in [3] that if (X,τ) is quasi-H-closed andᏵis an ideal such thatᏵn⊂Ᏽ, then (X,τ) is (Ᏽ) compact (and henceC(Ᏽ)-compact).
Next, if{U1,U2,...,Un}is a finite collection of open subsets such thatX−ni=1cl(Ui)∈
Ᏽn, thenX−ni=1cl(Ui)=φbecauseτ∩Ᏽn= {φ}. But then int(cl(X−
n
i=1Ui))=X−
n
i=1cl(Ui)=φimplies thatX−
n
i=1Ui∈Ᏽn. Therefore, a space (X,τ) is (Ᏽn) compact
if and only if it isC(Ᏽn)-compact. In view of this discussion, we have the following. Theorem 3.5. For a space (X,τ), the following are equivalent:
(a) (X,τ) is quasi-H-closed; (b) (X,τ) is (Ᏽn) QHC; (c) (X,τ) isC(Ᏽn)-compact; (d) (X,τ) is (Ᏽn) compact.
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 ifτ∩Ᏽm= {φ}, whereᏵmis the ideal of meager (first category) subsets of (X,τ). Thus, in view of the above theorem, a Baire space (X,τ) isC(Ᏽm)-compact if and only if it is quasi-H-closed.
We now give some characterizations ofC(Ᏽ)-compact spaces.
Theorem 3.6. Let (X,τ) be a space and let Ᏽbe an ideal onX. Then the following are equivalent:
(a) (X,τ) isC(Ᏽ)-compact;
(b) for each closed subset Aof X and each familyᏲ of closed subsets ofX such that {FA:F∈Ᏺ} =φ, there exists a finite subfamily {F1,F2,F3,...,Fn} such that
(int(Fi))∩A∈Ᏽ;
(c) for each closed setAand each familyᏲof closed sets such that{int(F)∩A:F∈Ᏺ} has (Ᏽ) FIP, one has∩{F∩A:F∈Ᏺ} =φ;
(d) for each closed setAand each regular open coverᐁofA, there exists a finite subcol-lection{U1,U2,U3,...,Un}such thatA−
n
i=1cl(Ui)∈Ᏽ;
(e) for each closed setAand each familyᏲof regular closed sets such that{F∩A:F∈
Ᏺ} =φ, there is a finite subfamily{F1,F2,F3,...,Fn}such thatni=1(int(Fi))∩A∈
Ᏽ;
(f) for each closed setAand each familyᏲof regular closed sets such that{int(F)∩A: F∈Ᏺ}has (Ᏽ) FIP, one has{F∩A:F∈Ᏺ} =φ;
(g) for each closed setA, each open coverᐁofX−Aand each open neighborhoodV ofA, there exists a finite subfamily {U1,U2,U3,...,Un} ofᐁsuch thatX−(V∪
(h) for each closed set A and each open filter baseᏮonXsuch that{B∩A:B∈Ꮾ} ⊂ P(X)−Ᏽ, one has{cl(B) :B∈Ꮾ} ∩A =φ.
Proof. (a)⇒(b). Let (X,τ) beC(Ᏽ)-compact,Aa closed subset, andᏲa family of closed subsets with∩{F∩A:F∈Ᏺ} =φ. Then{X−F:F∈Ᏺ}is an open cover ofA and hence admits a finite subfamily{X−Fi:i=1, 2,...,n}such thatA−ni=1cl(X−Fi)∈Ᏽ.
This set inᏵis easily seen to beni=1{int(Fi)∩A}.
(b)⇒(c). This is easy to be established.
(c)⇒(a). LetA be a closed subset, letᐁbe an open cover of Awith the property that for no finite subfamily{U1,U2,U3,...,Un}ofᐁ, one hasA−ni=1cl(Ui)∈Ᏽ. Then
{X−U:U∈ᐁ}is a family of closed sets. Since
n
i=1
X−clUi∩A= n
i=1
A−clUi=A− n
i=1
clUi, (3.4)
the family{int(X−U)∩A:U∈ᐁ}has (Ᏽ) FIP. By the hypothesis{(X−U)∩A:U∈
ᐁ} =φ. But thenA− ∪{U:U∈ᐁ} =φ, that is,ᐁis not a cover ofA, a contradiction. (d)⇒(a). LetAbe a closed subset ofXandᐁan open cover ofA. Then{int(cl(U)) : U∈ᐁ}is a regular open cover of A. Let{int(cl(Ui)) :i=1, 2,...,n} be a finite sub-family such thatA−ni=1cl(int(cl(Ui)))∈Ᏽ. SinceUi is open and for each open set
U, cl(int(cl(U)))=cl(U), we have A−ni=1cl(Ui)∈Ᏽ, which shows that X is C(Ᏽ
)-compact.
(a)⇒(d). This is obvious.
The proofs for (d)⇒(e)⇒(f)⇒(d) are parallel to (a)⇒(b)⇒(c)⇒(a), respectively. (a)⇒(g). LetAbe a closed set,Van open neighborhood ofA, andᐁan open cover of X−A. SinceX−V⊂X−A,ᐁis also an open cover of the closed setX−V.
Let{U1,U2,U3,...,Un}be a finite subcollection ofᐁsuch that (X−V)−
n
i=1cl(Ui)∈
Ᏽ. However, the last set isX−(V∪ {ni=1cl(Ui)}).
(g)⇒(a). LetAbe a closed subset ofXandᐁan open covering ofA. IfHdenotes the union of members ofᐁ, thenF=X−H is a closed set andX−Ais an open neighbor-hood ofF. Alsoᐁis an open cover ofX−F. By hypothesis, there is a finite subcollection {U1,U2,U3,...,Un}ofᐁsuch that
X
(X−A)∪ n
i=1
clUi
∈Ᏽ. (3.5)
However, this set inᏵis nothing butA−ni=1cl(Ui).
(a)⇒(h). SupposeAis a closed set andᏮis any open filter base onX with{B∩A: B∈Ꮾ} ⊂P(X)−Ᏽ. Suppose, if possible,{cl(B) :B∈Ꮾ} ∩A=φ. Then{X−cl(B) : B∈Ꮾ} is an open cover ofA. By the hypothesis, there exists a finite subfamily{X− cl(Bi) :i=1, 2, 3,...,n}such thatA−ni=1cl(X−cl(Bi)) is inᏵ. However, this set isA∩
(ni=1int(cl(Bi))) andA∩(
n
i=1Bi) is a subset of it. Therefore,A∩(
n
i=1Bi)∈Ᏽ. Since
Ꮾis a filter base, we have a B∈Ꮾsuch thatB⊂ni=1Bi. But thenA∩B∈Ᏽ which
contradicts the fact that{B∩A:B∈Ꮾ} ⊂P(X)−Ᏽ.
ofᐁ, we haveA−ni=1cl(Ui)∈/ Ᏽ. We may assume thatᐁis closed under finite unions.
Then the familyᏮ= {X−cl(U) :U∈ᐁ}is an open filter base onXsuch that{B∩A: B∈Ꮾ} ⊂P(A)−Ᏽ. So, by the hypothesis,{cl(X−cl(U)) :U∈ᐁ} ∩A =φ. Letxbe a point in the intersection. Thenx∈Aandx∈cl(X−cl(U))=X−int(cl(U))⊂X−U for eachU∈ᐁ. But this contradicts the fact thatᐁis a cover ofA. Hence (X,τ) isC(Ᏽ
)-compact.
Next we characterizeC(Ᏽ)-compact spaces using some weaker forms of filter base con-vergence.
Definition 3.7. A filter baseᏮis said to be (Ᏽ) adherent convergent if for every neighbor-hoodGof the adherent set ofᏮ, there exists an elementB∈Ꮾsuch that (X−G)∩B∈Ᏽ. 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) isC(Ᏽ)-compact if and only if every open filter base onP(X)−
Ᏽis (Ᏽ) adherent convergent.
Proof. Let (X,τ) beC(Ᏽ)-compact and letᏮbe an open filter base onP(X)−Ᏽwith A as its adherent set. LetG be an open neighborhood of A. ThenA={cl(B) :B∈
Ꮾ},A⊂G, andX−G is closed. Now{X−cl(B) :B∈Ꮾ} is an open cover ofX−G and so by the hypothesis, it admits a finite subfamily{X−cl(Bi) :i=1, 2, 3,...,n}such that (X−G)−ni=1cl(X−cl(Bi))∈Ᏽ. But this implies (X−G)∩(
n
i=1int(cl(Bi)))∈Ᏽ.
However,Bi⊂int(cl(Bi)) implies (X−G)∩(ni=1Bi)∈Ᏽ. SinceᏮis a filter base and
Bi∈Ꮾ, there is aB∈Ꮾsuch thatB⊂in=1Bi. But then (X−G)∩B∈Ᏽis required.
Conversely, let (X,τ) be notC(Ᏽ)-compact, and letAbe a closed set, andᐁan open cover ofAsuch that for no finite subfamily{U1,U2,U3,...,Un}ofᐁ, one hasA−ni=1
cl(Ui)∈Ᏽ. Without loss of generality, we may assume thatᐁis closed for finite unions. Therefore,Ꮾ= {X−cl(U) :U∈ᐁ} becomes an open filter base onP(X)−Ᏽ. If x is an adherent point ofᏮ, that is, ifx∈{cl(X−cl(U)) :U∈ᐁ} =X−{int(cl(U)) : U∈ᐁ}, thenx /∈A, becauseᐁis an open cover ofAand forU∈ᐁ,U⊂int(cl(U)). Therefore, the adherent set of Ꮾis contained in X−A, which is an open set. By the hypothesis, there exists an elementB∈Ꮾsuch that (X−(X−A))∩B∈Ᏽ, that is,A∩ B∈Ᏽ, that is,A∩(X−cl(U))∈Ᏽ, that is,A−cl(U)∈Ᏽfor someU∈ᐁ. This however
contradicts our assumption. This completes the proof.
Herrington and Long [4] characterizedC-compact spaces usingr-convergence of fil-ters and nets. We obtain similar results forC(Ᏽ)-compact spaces in the next definition.
Definition 3.9. LetX be a space,φ =A⊂X, and letᏮbe a filter base onA.Ꮾis said tor-converge toa∈A if for each open setV in X containinga, there is B∈Ꮾwith B⊂cl(V). The filter baseᏮis said tor-accumulate toa, if for each open setVcontaining a, cl(V)∩B =φfor eachB∈Ꮾ.
It is known [4] that convergence (accumulation) for filter bases and nets implies r-convergence (r-accumulation), but the converse is not true.
Theorem 3.10. For a space (X,τ) and an idealᏵonX, the following are equivalent: (a) (X,τ) isC(Ᏽ)-compact;
(b) for each closed setA, each filter baseᏮonP(A)−Ᏽr-accumulates to somea∈A; (c) for each closed setA, each maximal filter baseᏹonP(A)−Ᏽr-converges to some
a∈A;
(d) for each closed setA, each netϕonP(A)−Ᏽr-accumulates to somea∈A.
Proof. (a)⇒(b). Suppose there exist a closed setAand a filter baseᏮonP(A)−Ᏽwhich does notr-accumulate to anya∈A. Then for eacha∈A, there exists an open setU(a) containingaand aB(a)∈Ꮾsuch thatB(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(ai) :i=
1, 2, 3,...,n}such thatA−ni=1cl(U(ai))∈Ᏽ. IfB∈Ꮾis such thatB⊂
n
i=1B(ai), then
B∩(A−ni=1cl(U(ai)))∈Ᏽ, that is,B−
n
i=1cl(U(ai))∈Ᏽ. But the later set is justB,
be-causeB⊂B(ai) andB(ai)∩cl(U(ai))=φfor eachi. However,B∈Ᏽis a contradiction,
becauseB∈ᏮandᏮ⊂P(A)−Ᏽ.
(b)⇔(c). This follows in view of parts (a), (b), and (c) of [4, Theorem 1].
(b)⇒(a). If possible, letXbe notC(Ᏽ)-compact. Then byTheorem 3.6(f), there exist a closed setAand a collectionᏲof regular closed sets with the property that for every finite subcollection{F1,F2,F3,...,Fn},ni=1int(Fi)∩A /∈Ᏽ, but
{F:F∈Ᏺ} ∩A=φ. Now the collection of sets of the form ni=1int(Fi)∩Afor all possible finite
subfami-lies{F1,F2,F3,...,Fn} of Ᏺ forms a filter base onP(A)−Ᏽ. By (b), this filter base
r-accumulates to somea∈A, that is, for each open setU(a) containingaand for each F∈Ᏺ, cl(U(a))∩(int(F)∩A) =φ. However,a∈AandA∩ {F:F∈Ᏺ} =φimply that there is someF=F(a)∈Ᏺsuch thata /∈F(a). ThenX−F(a) is an open set containing asuch 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,Ais replaced by the whole spaceX, we get the characteriza-tions of (Ᏽ) QHC spaces. If in addition we consider completely codense idealᏵ, we get the characterizations of quasi-H-closed spaces.
4.C(Ᏽ)-compact spaces and functions
A function f : (X,τ)−(Y,σ) is said to beθ-continuous [2] at a pointx∈X if for ev-ery open setV ofY containing f(x), there exists an open setUofXcontainingxsuch that f(cl(U))⊆cl(V). A function f : (X,τ)−(Y,σ) is said to beθ-continuous if f is θ-continuous for every x∈X. The concept ofθ-continuity is weaker than that of conti-nuity. An important property ofC-compact spaces is that a continuous function from aC-compact space to a Hausdorffspace is closed. We prove the following more general results.
Proof. LetAbe any closed set inXanda /∈ f(A). For eachx∈A, there exists aσ-open setVycontaining y= f(x) such thata /∈cl(Vy). Now because f isθ-continuous, there
exists an open setUx containingx such that f(cl(Ux))⊆cl(Vy). The family {Ux:x∈ A}is an open cover ofA. Therefore, there exists a finite subfamily{Uxi:i=1, 2,...,n}
such thatA−ni=1cl(Uxi)∈Ᏽ. But then f(A−
n
i=1cl(Uxi))∈f(Ᏽ)⊆ϑ, that is, f(A)−
f(ni=1cl(Uxi))∈ f(Ᏽ)⊆ϑbecause f(Ᏽ) is also an ideal. Hence f(A)−(
n
i=1cl(Vyi))∈
f(Ᏽ)⊆ϑ. Nowa /∈cl(Vyi) for anyiimplies thata∈Y−
n
i=1cl(Vyi) which is open in
(Y,σ) and (Y−ni=1cl(Vyi))∩f(A)=f(A)−
n
i=1cl(Vyi)∈ f(Ᏽ)⊆ϑ. Hencea /∈(f(A))∗
(σ,ϑ). Thus (f(A))∗(σ,ϑ)⊂f(A) and so f(A) isσ∗(ϑ)-closed. Corollary 4.2. Let f : (X,τ,Ᏽ)−(Y,σ,ϑ) be a continuous function, (X,τ,Ᏽ)C(Ᏽ )-com-pact, (Y,σ) Hausdorff, and f(Ᏽ)⊆ϑ. Then f(A) isσ∗(ϑ)-closed for each closed setAof X.
Theorem 4.3. Let f : (X,τ,Ᏽ)−(Y,σ,ϑ) be a continuous surjection, (X,τ,Ᏽ)C(Ᏽ )-com-pact, and f(Ᏽ)⊆ϑ. Then (Y,σ,ϑ) isC(ϑ)-compact.
Proof. LetAbe any closed subset of (Y,σ) and{Vα:α∈Λ}any open cover ofAby open sets inY. Then{f−1(V
α) :α∈Λ}is an open cover off−1(A) which is closed inX. Hence,
by the hypothesis, there exists a finite subcollection{f−1(V
αi) :i=1, 2,...,n}such that
f−1(A)−n
i=1cl(f−1(Vαi))∈Ᏽ. Sincef is continuous, cl(f−1(B))⊂ f−1(cl(B)) for every
subsetBofY. Hence we have f−1(A)−n
i=1f−1(cl(Vαi))=f−1(A−
n
i=1cl(Vαi))∈Ᏽ.
Since f is surjective,A−ni=1cl(Vαi)∈ f(Ᏽ)⊂ϑ. HenceY isC(ϑ)-compact.
Theorem 4.4. If the product spaceΠXαof nonempty family of topological spaces (Xα,τα) is
C(Ᏽ)-compact, then each (Xα,τα) isC(pα(Ᏽ))-compact, wherepαis the projection map and Ᏽis an ideal onΠXα.
Proof. This follows fromTheorem 4.3.
5.C(Ᏽ)-compact spaces and subspaces
In this section, we introduce three types ofC(Ᏽ)-compact subsets and use them to obtain new characterizations ofC(Ᏽ)-compact spaces and a characterization of maximalC(Ᏽ )-compact spaces.
Definition 5.1. Let (X,τ) be a space andᏵan ideal onX. A subsetY ofX is said to be C(Ᏽ)-compact if the subspace (Y,τY) isC(Ᏽ)-compact.
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 andᏵan ideal onX. Then
(a) a subspaceY isC(Ᏽ)-compact if and only if it isC(ᏵY)-compact; (b) a clopen subspace of aC(Ᏽ)-compact space isC(Ᏽ)-compact;
(c) if Y is a regular closed subset of aC(Ᏽ)-compact space (X,τ,Ᏽ) andᏵis codense, then (Y,τY) is quasi-H-closed;
Definition 5.3. A subsetYof (X,τ) is said to beC(Ᏽ)-compact relative toτif everyτ-open cover of every relatively closed subsetAofYhas a finite subfamily whoseτ-closures cover Aexcept a set inᏵ.
Some useful properties of such spaces are contained in the following.
Theorem 5.4. Let (X,τ) be a space andᏵan ideal onX. Then the following hold.
(a) A closed subspace of aC(Ᏽ)-compact relative toτsubspace of (X,τ) isC(Ᏽ)-compact relative toτ.
(b) If (X,τ) is HausdorffandY isC(Ᏽ)-compact relative toτ, thenY isτ∗(Ᏽ)-closed. (c) If Y is a C(Ᏽ)-compact relative toτsubspace of (X,τ) and f : (X,τ)−(Z,σ) is a
continuous bijection, thenf(Y) isC(f(Ᏽ))-compact relative toσ. (d)C(Ᏽ)-compactness relative toτis contractive.
The following characterization ofC(Ᏽ)-compact spaces is obtained usingC(Ᏽ )-com-pact relative toτsubspaces. The proof is easy.
Theorem 5.5. A space (X,τ) with an idealᏵisC(Ᏽ)-compact if and only if every proper closed subset ofXisC(Ᏽ)-compact relative toτ.
Definition 5.6. AsubsetYof a space (X,τ) is said to be closureC(Ᏽ)-compact if for every τY-closed subsetKofYand everyτ-open coverᐁof cl(K), there is a finite subcollection {U1,U2,U3,...,Un}ofᐁsuch thatK−
n
i=1clY(Ui∩Y)∈Ᏽ.
It is easy to see that closureC(Ᏽ)-compactness is contractive.
Example 5.7. Since closed subsets ofC(Ᏽ)-compact spaces are not necessarily (Ᏽ) QHC, a space (X,τ) which isC(Ᏽ)-compact relative toτmay fail to be closureC(Ᏽ)-compact. Moreover, ]0, 1] as a subspace of [0, 1] is closureC(Ᏽ)-compact withᏵ= {φ}, but not C(Ᏽ)-compact relative to the usual topology. Thus the concepts ofC(Ᏽ)-compact relative toτand closureC(Ᏽ)-compact are independent concepts.
We now have the following characterization ofC(Ᏽ)-compact spaces.
Theorem 5.8. A space (X,τ) isC(Ᏽ)-compact for an idealᏵonXif and only if every open subset ofXis closureC(Ᏽ)-compact.
Proof. Let (X,τ) beC(Ᏽ)-compact andY an open subset ofX. Let K be aτY-closed
subset ofY, and letᐁbe aτ-open cover of cl(K). Then there exists a finite subcollec-tion{U1,U2,U3,...,Un}ofᐁsuch that clK−
n
i=1cl(Ui)∈Ᏽ. SinceYis open, therefore,
clY(U∩Y)=cl(U)∩Y and so, by hereditary property ofᏵ,K−ni=1clY(Ui∩Y)∈Ᏽ.
ThusYis closureC(Ᏽ)-compact.
Conversely, let all open subsets ofXbe closureC(Ᏽ)-compact. LetK be a closed and
ᐁan open cover ofK. Choose aU0∈ᐁ. ThenY=X−cl(U0) is an open subset ofXand
K∩Y is aτY-closed subset ofY. Moreover,ᐁ− {U0}is an open cover of cl(K∩Y). By
the hypothesis, there exists a finite subcollection{U1,U2,U3,...,Un}ofᐁ− {U0}such
thatK∩Y−ni=1clY(Ui∩Y)∈Ᏽ. But thenK∩Y−
n
i=1cl(Ui)∈Ᏽas clY(Ui∩Y)=
cl(Ui)∩Y andᏵ is hereditary. Therefore, K−in=0cl(Ui)∈Ᏽ. Hence (X,τ) is C(Ᏽ
Finally, we obtain a characterization of a maximalC(Ᏽ)-compact space. Recall that a space (X,τ) with propertyPis said to be maximalPif there is no topologyσonXwhich has propertyP and is strictly finer thanτ. For a topological space (X,τ) and a subsetA ofX,τ(A)= {U∪(V∩A) :U,V∈τ}is a topology called simple extension [7] ofτbyA. τ(A) is strictly finer thanτif and only ifA /∈τ. Theorem 5.9. A topological space (X,τ) is maximalC(Ᏽ)-compact if and only if for every subsetAofXsuch thatAis closureC(Ᏽ)-compact andX−AisC(Ᏽ)-compact relative toτ, one hasA∈τ.
Proof. First we assume that (X,τ) is maximalC(Ᏽ)-compact and thatAis a subset ofX satisfying the given conditions. First, we show that (X,τ(A)) isC(Ᏽ)-compact. LetK be aτ(A)-closed subset ofX. ThenK=K1∪(K2∩(X−A)), whereK1andK2areτ-closed
sets. Let
ᐁ=Uα∪Vα∩A:Uα,Vα∈τ,α∈Δ (5.1)
be aτ(A)-open cover ofK. Thenν= {Uα:α∈Δ}is aτ-open cover ofK∩(X−A)=
(K1∪K2)∩(X−A). Since, by assumption,X−AisC(Ᏽ)-compact relative toτ, we have a
finite subcollection{Uα1,Uα2,Uα3,...,Uαn}ofνsuch thatK∩(X−A)−
n
i=1cl(Uαi)∈Ᏽ.
Since τ(A) is finer than τ, this subcollection is τ(A)-open and K∩(X −A)− n
i=1clτ(A)(Uαi)∈Ᏽ. Next, ᐃ= {Uα∪Vα:α∈Δ} is a τ-open cover of cl(K∩A)=
cl(K1∩A)=clτ(A)(K1∩A) and therefore by assumption onA, there exists a finite
sub-collection{Uβi∪Vβi:i=1, 2,...,k}ofᐃsuch that
K1∩A−
k
i=1
clτA
Uβi∪Vβi
∩A∈Ᏽ. (5.2)
However,τA, the restriction ofτtoA, is nothing butτ(A)|A, the restriction ofτ(A) to
A. Therefore,
K1∩A−
k
i=1
clτ(A)|AUβi∪Vβi
∩A∈Ᏽ. (5.3)
Now{Uαi∪(Vαi∩A) :i=1, 2,...,n} ∪ {Uβi∪(Vβi∩A) :i=1, 2,...,k}is a finite τ(A)
(Ᏽ) proximate cover ofK which is a subcover ofᐁ. Thus the topologyτ(A) onXis also C(Ᏽ)-compact. However, by the maximality ofτ, we haveτ(A)=τ. But thenA∈τas desired.
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M. K. Gupta: Department of Mathematics, Faculty of Science, Ch. Charan Singh University, Meerut-250004, India
E-mail address:[email protected]
