NEW ZEALAND JOURNAL OF MATHEMATICSVolume 22 (1993), 63-66

A N O TE C H A R A C T E R IZ IN G CO U N TA B LE C O M PACTN ESS W IT H R E SP E C T TO A N ID EAL

T . R . H a m l e t t a n d D a v i d R o s e

(Received July 1991)

Abstract. An ideal is a nonempty collection of subsets of a space closed under the operations of subset and finite union. A space X is said to be (countably) compact with respect to X, or simply (countably) X-compact, if every (countable) open cover of the space admits a finite subcollection which covers all the space except for a set in the ideal. It is shown that a space X with an ideal I is countably Z-compact if and only if every locally finite collection of non-ideal subsets is finite. A space X with ideal X is said to be paracompact with respect to I , or simply T-paracompact, if every open cover of the space admits a locally finite open refinement (not necessarily a cover) which covers all the space except for a set in the ideal. Let ( X , r , l ) be a space with ideal 1 such that X is I-paracompact and I n t — { 0 } . It is shown that in this setting, countable X-compactness and I-compactness are equivalent. Special cases include: countable compactness is equivalent to compactness in paracompact spaces; light compactness is equivalent to quasi //-closedness in almost paracompact spaces; and countable meager-compactness is equivalent to meager-compactness in meager-paracompact Baire spaces.

An ideal is a nonempty collection of subsets of a space X closed under the operations of subset (heredity) and finite union (finite additivity). An ideal closed under the operation of a countable union is called a cr-ideal. If X is not a member of the ideal, then the ideal is said to be proper. Observe that the collection of complements of a proper ideal form a filter, hence proper ideals are also called dual filters.

We will denote by (X , r ,J ) a topological space (X , r) and an ideal 1 on X . If A C X , then we denote by C1(A) and Int(.A) the closure and interior of A, respectively. If x e X , then we denote the open neighborhood system at x by t (x \,i.e., r ( x) = { U £ r : x E U } . We denote the natural numbers by IN. We abbreviate the phrase “if and only if” by “iff” .

A subset A C X i s said to be a non-ideal set if A £ X.

The concept of “countable compactness modulo an ideal J ” or “countable J- compactness” was introduced by Newcomb in [1] and has been further studied by Hamlett, Jankovic, and Rose in [2]. For definitions and basic facts related to this concept and the special case of light compactness, the reader is referred to [2].

A collection A of subsets of a space X is said to be locally finite if for every x G X there exists a U G r(x) such that U has a non empty intersection with at most finitely many members of A . It is known that a space is countably compact iff locally finite collections of (nonempty) subsets are finite, and it is known that a space is lightly compact iff locally finite collections of (nonempty) open subsets are finite [3]. We prove in this paper a theorem which has both of these known facts as corollaries.

1991 AMS Mathematics Subject Classification: 54D30, 54D20

64 T.R. HAMLETT AND DAVID ROSE

Theorem 1. Let ( X ,t,X) be a space. Then (X ,r ) is countably X-compact iff every locally finite collection of non-ideal sets is finite.

P roof. Necessity. Assume that (X ,r ) is countably X-compact and {A n : n e IN} is a countably infinite locally finite family of non-ideal sets. For every n G IN, define Bn = U^ nAi. Observe that Bi D B 2 D B 3 . . . and that Bn £ X for every n. Let x G X , then there exists U G t{x ) and j G IN such that U fl Ai = 0 for everyi > j . Hence U n Bj = 0 and x £ C\(Bj), showing that D^L1cl(Sn) = 0 . Thus { X — Cl(Bn) : n G IN} is a countable open cover of X and hence there exists a finite subcollection { X — Cl(Bni) : i = 1 ,... ,ra} such that X — (X — Cl(Bni)) = n^.1Cl(Bni) G X. Choosing some natural number K > max{ni, 712, . . . ,n m}, we have B k C n-^1S Tli C f l^ 1Cl(Bni) which implies B k G X by heredity, and this is a contradiction.

Sufficiency. We proceed to show the contrapositive. Assume (X, r) is not count­ably X-compact. Then there exists a countable open cover {Un : n G IN} such that for every finite set F C IN, X — U{Uk : k G F } £ X. Let A n — X — U{Uk : k = 1 ,2 ,... , n}. Then A\ D A 2 D A 3 . . . , each A n £ J, and n ^ 1yln = 0 . Let x G X , then since each A n is closed there exists U G t(x ) and m G IN such that U n A n = 0 for every n > m. Since x was arbitrary, we conclude that {A n : n G IN} is a locally finite family. If {A n : n G IN} was a finite collection of sets, then we would have for some natural number K , A n = A m for every n, m > K . This would simply imply

= A k 7̂ 0 which is a contradiction. Hence {A n : n G IN} is a locally finite infinite collection of non-ideal sets. I

Letting X = { 0 } in the above theorem, we obtain the following well known result.

Corollary 1. A space is countably compact iff every locally finite collection of nonempty sets is finite. I

A subset of a space is said to be somewhere dense if it is not nowhere dense. Letting X be the ideal of nowhere dense sets in the previous theorem, we obtain the following interesting characterization of light compactness.

Corollary 2. A space is lightly compact if (1) every locally finite collection of somewhere dense sets is finite. I

It is well known [3] that a space is lightly compact iff (2) every locally finite collection of (nonempty) open sets is finite. It is clear that (1) implies (2) since nonempty open sets are somewhere dense. It is left to the interested reader to show(2) implies (1) directly.

A space is said to be paracompact if every open cover of the space has a locally finite open refinement which covers the space (we do not require a refinement to be a cover). The well known result that countable compactness is equivalent to compactness in paracompact spaces follows immediately from Corollary 1. A space is said to be quasi H-closed [8], abbreviated QHC, if every open cover of the space has a finite subcollection whose union is dense in the space. A QHC Hausdorff space is said to be H-closed [9]. A space is said to be almost paracompact [4] if every open cover of the space has a locally finite open refinement whose union is dense

A NOTE CHARACTERIZING COUNTABLE COMPACTNESS WITH RESPECT TO AN IDEAL 65

in the space, or, equivalently, if every open cover of the space has a locally finite open refinement such that the closures of its members cover the space. A Hausdorff almost paracompact space is said to be para-i/-closed [3]. The following result of Zahid [3], then follows immediately from condition (2): if-closedness is equivalent to light compactness in para-//-closed spaces. The following result improves this result of Zahid by eliminating the Hausdorff assumption.

Theorem 2. Light compactness is equivalent to quasi H-closedness in almost paracompact spaces.

P roof. We need only show that an almost paracompact lightly compact space is QHC. Let X be an almost paracompact lightly compact space, and let U be an open cover of X . Since X is almost paracompact, U has a locally finite open refinementV such that UV is dense in X . We may assume without loss of generality that each member of V is nonempty and hence somewhere dense. It follows from Corollary2 that V is finite, and hence X is QHC. I

A space (X ,r ,X ) is said to be X-compact [1] [5] [6] if for every open cover {Ua : ol G A } of X , there exists a finite subcollection {Uai : i = 1 ,2 ,... , n } such that X — U?=1Uai G X. We say a space X is X-paracompact ( “paracompact modulo X” in [3]) if every open cover U of X has a locally finite open refinement V such that X — UV G X. With these definitions, we can state the following general theorem.

Theorem 3. Let ( X ,t ,X) be an X-paracompact space with 1 0 r = { 0 } . Then (X, r) is countably X-compact iff ( X ,t ) is X-compact.

P roof. We need only show necessity. Assume (X ,t , X) is a countably J-compact X-paracompact space, and let U be an open cover of X . Since X is X-paracompact, U has a locally finite refinement V of nonempty open sets such that X — UV G X. Since X is countably X-compact and X D r = { 0 }, it follows from Theorem 1 thatV is finite, and hence X is X-compact. I

It is well known [7] that a space (X, t ) is a Baire space iff M {t)C \t = { 0 }, where M (r ) denotes the cr-ideal of meager (first category) subsets of X . The following corollary follows immediately.

Corollary 3. Let (X , r) be a Baire M ( t )-paracompact space. Then X is countably M ( t )-compact iff X is M(r)-compact. I

Other such corollaries can be stated simply by choosing an appropriate ideal. For example, the ideal of subsets of Haar measure zero in a locally compact topological group has the desired property that nonempty sets of Haar measure zero have empty interior. More specifically, the ideal of subsets of the usual real line of Lebesgue measure zero has the desired property.

66 T.R. HAMLETT AND DAVID ROSE

References

1. R.L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. disser­tation, Univ. of. Cal. at Santa Barbara, 1967.

2. T.R. Hamlett, D. Jankovic and D.R. Rose, Countable compactness with respect to an ideal, Math. Chronicle 20 (1991), 109-126.

3. M.I. Zahid, Para-H-closed spaces, locally para-H-closed spaces and their min­imal topologies, Ph.D. dissertation, Univ. of Pittsburgh.

4. M.K. Singal and S.P. Arya, On m-paracompact spaces, Math. Ann. 181 (1969), 119-133.

5. T.R. Hamlett and D. Jankovic, Compactness with respect to an ideal, Boll. U.M.I. (7) 4-B (1990), 849-862.

6. D.V. Rancin, Compactness modulo an ideal, Soviet Math. Dokl. 13 No. 1 (1972), 193-197.

7. R.C. Haworth and R.A. McCoy, Baire spaces, Diss. Math., CXLI.8. H.V. Velicko, H-closed topological spaces, Mat. Sb. (N.S.) 70 (112) (1966),

98-112; Amer. Math. Soc. Transl. (2) 78 (1969), 103-118.9. M.P. Berri, J.R. Porter and R.M. Stephenson (Jr), A survey of minimal topo­

logical spaces, Proc. Kanpur Topological Conference (1968), in General Topol­ogy and Its Relations to Modem Analysis and Alqebra III , Academic Press, New York, 1970.

T.R . Hamlett and David Rose East Central University AdaOklahoma 74820 U.S.A