Kyungpook M a.thematical Journ 매 Volume 32, Nurnber 1, June, 1992

LOCAL COMPACTNESS WITH RESPECT TO AN IDEAL

T.R. Hamlett and David Rose

Given a nonempty set X , an ideal I on X is a collection of subsets operatl0%moduloof X closedan ideal under”, Sovietfin 따 unionMath .and DOM. subset Vol.l3 (1972), No.1) Raniin stud ied a generalization of compactness (I-compact야 ss) which requires that open covers of a space have a finite subcollection which covers all the space except for a set in the ideal. In this paper we define a space to be locally I-compact if every point in the space has an I-compact neighbor­ hood. Basic results concerning locally I -compact spaces are given relating to subspaces, preservation by functions, and products. Classical results concerning locally ∞mpact spaces are obtained by letting I = {0 }, and certain results for locally H -closed spaces are obtained by letting I be t he ideal of nowhere dense sets. Locally H -closed spaces are characterized in the category of Hausdorff spaces as being the locally nowhere-dense compact spaces

1. Introduction Given a nonempty set X , a collection I of subsets of X is called an ideal if (1) A E I and B 드 A implies B E I (heredity), and (2) A E I and B E I implies A U B E I (finite additivity). If X rf. I then I is called a proper ideal. Note that if I is a proper ideal then {A : X - A E I} is a 5lter; hence, proper ideals are sometimes called d ual fil ters

R.eceived October 28, 1990. This research was partially supported by a graot from East Ce otral U niversity Keywords and Phrases: !deal, compact, I -compact, locally compact, locally I compaιt ， continuous function , open function , pointwise I -continuous fun ction , point­ wise I -open fun ction , nowhere dense sets, qu 앙 H-cl。잊 d spaι05 ， H-closed sp값es ， locally H-closed spaces 1980 AMS Classification Codes. 54D30, 51D45

31 32 T.R. Hamlett and David Rose

We denote by (X,r,T) a topological space (X,r) and an ideal T of subsets of X. Consider the following definition.

Definition. [9][13][3] A subset A of a space (X, r ,T ) is said to be T­

compact if for every open cover {U", : a E [1} of A, there exists a fin 따 sub­ collection {U'" ‘ : i = 1,2,"', n} such that A-U{U"" : i = 1,2,"', n} E T. The space (X, r,T) is said to be T- compact if X is T-compact as a subset.

A subset A of a space (X, r ) is said to be bounded [8] iff every open cover of X contains a fini te subcollection wh ich covers A. If 5 ç X , we denote by < S >= {B 드 X:B 드 S}, the principal ideal generated by S. Observe that a subset A of a space (X, r ) is bounded iff X is < X - A > compact. We see then that compactness with respect to an ideal is useful as a unifi cation and generalization of several other widely studied concepts. Compactness with respect to an ideal (T- compactness) has been stlld ied extensively in [3] , [13] , and [9]. A topological space (X π) is compacl in the l1sual sense iff (i“f and 이아떼0 n비llyy 川iff) “it 염i s {Ø깨} -c이oorr매 ac야t ’. w‘Io/e say a space is quasi H-closed (QHC) iff every open cover of the space has a finite slIbcoJ Jection whose c10sures cover tbe space. A QHC T2 space is called H-c/os ed. T he following theorem is stated for reference

Theorem 1.1. Lel (X, r) be a topological space and lel N(r) denote the ideal of nowhere dense subsets of (X, r ). (1) [3] (X,r) isN(r)-compact 퍼 (X ， r) is QHC

(2) [1 3]lf (X,r) is T2 , then (X,r) isN(r)-compact iff(X,r) is H­ closed. In t his paper we study the naturaJ ex tension of the concept of com pactness with respect to an ideal to “Jocal" compactness with respect to an ideal ‘ Given a topoJogical space (X ,r) and x E X , we denote by r(x) {U E r : x E U} . A sllbset A of X is called a neighborhood, abbreviated “nbd" , of x if there exists U E r (x) such that x E U 드 A. Given a subset B of X , we denote by CI(B ) and Int(B) the c10sure and interior of B with respect to r , respectively. Given a space (X, r ,T ) and a subset A of X , we denote by A*(T, r ) = {x E X : U n A if. T for every U E r (x) }, the local functioη of A with πsp ect to T and r [15]. When no ambiguity is present, we simply write Local compactness with respect to an ideal 33

A" for A*(T ,T). The set operator C I*(A) = A u A* defì. nes a Kuratowski closure operator for a topology T*(I) fìner (Iarger) than T. A basis ß(T , T) for T"(T) can be described as ß(T , T) = {U - 1 : U E T,I E T}[16] . We write T* for T"(T) and ß for ß(T , T) when no ambiguity is present. We denote by Int*(A) the interior of A with respect to T‘ We use the symbol “•" to mean “implies" “this implies", or “wh ich implies" as fits the context, as well as function designation.

2. Local I-compactness Let us begin with the following defì. nition

Definition. A space (X, T,T) is said to be loca/ly T-compacl if every point in X has an T-compad nbd. The space is said to be strongly loca /lν T-compact if every point in X has a nbd base of T- compact sets. A subset A of X is said to be (strongly) locally T-compaci if (A ,TIA ,TIA) is (strongly) locall y 피 A-com p act 、애 er e TIA is the usual subspace to이po뼈) and 피T IA = {I n A : 1 E T}. It is readily seen that all T-compact space is locally T-compact and that a strongly locall y T-compact space is lo call y T-compact. We will see shortly, however, that an T-compact space (and hence a lo cally T-compact space) need not be strongly locally T-compact, even if the space is T2 ‘ Given a space (X, T, T), we say that T is regular ψith respect 10 T" if each point in X has a T-nbd base of T* -closed sets

Theorem 2.1. Let (X, T,T) be a T2 space. lf (X, T,T) is slrongly locally T-compact, then T is regular with respect to T* Proof Let X E X and U E T(X). Since (X, T,T ) is strongly locally T­ compact, there exists an T-compact subset F and W E T(X) such that

X E W 드 F 드 U. Since (X, T ) is T2 , F is T"-closed [9], and the proof is complete An ideal T on a space (X, T) is said to be T-bounda711 [9] iff T n T = {0}.

Theorem 2.2. Lel (X,T,T ), be a s띠pa αce 11ψv뼈j regular ilJ T is regular w뼈l Proηof Necess잉íty ís 이0 b~、v씨r이i o u s . To show suflì ciency, assume T is regular with respect to T" and let X E X , U E T(X) . Then there ex ists V E T"(X) such that X E V 드 Cl’ (V) 드 U. By Theorem 9 of [14] , there exists G E T(X) 34 T.R. Hamlett and Oavid Rose such that x E V 드 G 드 C/(G) 드 C/"(V) 드 U. Hence X has a T-nbd base of T-closed sets at each point and is regular. We note that if (X, T,I) is a space and I n T = {0}, then T and T" have the same regu/ar open su bsets (U 드 (X, T) is regular open iff U = Int(CI(U))) . Specifically, if U E T", then CI(U) = CI"(U). This follows easi ly from T heorem 9 of [14]. The follow ing example shows that local T-compactness does not imply strong local T-compactness.

Example 1. Let X = [0,1] be the closed unit interval with a the usual topology as a subspace of the reals, and let S = {* n 1,2,3,"'} Denote by < S >= {B 드 X:B 드 S}, the p,-incipal i d e미 generaled by S. Letting T = a"(T) , we have that T"{ T ) = T (in general, T*' = T" [10 ]). (X, a) is compact 좌 and henæ < S >-compact T2 • (X,T) is < S >­ compact T2 [9]. Observing that T is not regular (hence not regular with respect to T* ) we conclude from the previous theorem that (X , T) is not strongly locally T-compact.

We see then that local T-compactness and strong local T-compactness are not equivalent, even in T2 spaces, in contrast to the situation with regard to local {까 compactness (i.e. , the us ual ‘“%‘ s야tro띠Jng local {애”띠} -co mψpa야ctnes잃s wh비11κch are 여e qUJv뻐a떠le히따n1πt in T2 얘s p ace얹s. In 매t hi녕s pape야r 、we 、씨‘w"i비l11 concentrate on the conκcep마t 이0 f local T- comψpactness. \Ve re­ mark, h 아，veve r ， that if (X, T ,T) is I-compact and T is regular with res pect to T ‘, then (X , T ,T) is strongly locally T-compact since T*-closed subsets are T-compact [3 , Theorem 2.4] The usual notion of loca l compactness ([6] every point has a compact nbd) is obviously equ ivalent to local {에 -compac tness. In [12] Porter de­ fin es a Hausdorff space to be locally H -c/osed iff every point has a nbd which is H-closed as a s1!bspace. We define a space to be locally QHC if every point has a nbd which is QHC as a subspace and note that a

T2 space is locally H-closed iff it is locally QHC. The follow ing theo­ rems show that in T2 spaces the concepts of locall y H -closed and locally N(T ) 一 compact are eq uivalent If (X, T ,T) is a space and I< 드 X , we denote by I II< = {I n I< : 1 E T} = {I E T: 1 드 I<}. TI I< is easily seen to be an ideal on X.

Theorem 2.3. Lel (X, T,T) be a space with A 드 K 드 X. Then A is T !K - compact in (I< , T!K) (where TI I< 잉 lhe 1!s1!al lopology on I< as a subspace Local compactness with respect to an ideal 35 of(X,r)) iff A is I-compact in (X,r).

Proof (Necess따). Assume A is IJI<-compact in I< and let {U" : ü' E .6} be a r -open cover of A. Then {U" n I< : ü' E .6} is a rJ I< open cover of A and hence there exists a finite 5 뼈 collection {U"‘ n I< : i = 1,2, ... , n} such that A - U{U", n I<: i = 1,2,'" ,n} E IJ I<. The identity (1) A - U(U", n I<) = A - UU", shows that A is I-compact (Sufficiency). Assume A is I-compact and let {U" n I< : ü' E .6} be a rJI<-open cover of A. Then {U" Ü' E .6} is a r-open cover of A Extracting a finite subcollection which covers all of A except for a part of A in the ideal and using the above identity (1) complet얹 the proof

Corollary 2.4. Let (X,r , I) be a space with F 드 X an I-compact subset. [f A 드 F is r’ closed, the η A is I-compact. Proof It is easily shown that (rJF)*(IJF) = r‘ (I) JF. Thus A being r*­ closed • A is IJF- compact in the subspace F • A is I-compact by the previous theorem The well known result that closed subsets of compact subsets are com pact follows from Corollary 2.4 by taking I = {Ø}

Theorem 2.5. Let (X,r ,I) be a 자 space. Then (X, r) is locally I compact iff for every x E X there exists U E r(x) such that CI*(U) is I-compact Proof Sufficiency is obvious. To show necessity, assume (X, r ,I) is locally I-compact and let x E X. There exists U E r(x) and an I-compact F 드 X such that x E U 드 F. F is r 까 losed [9] since (X, r) is T2 , and hence Cl‘ (U) 드 F. But Cl‘ (U) is a r‘ closed su bset of an I- cor매 act subset, and hence I-compact, by Corollary 2.4.

、‘Te will need the following lemma to establish the equivalence of local

N( r )-compactness and local H-closedness in T2 spaces Recall that a subset F of a space (X, r) is said to be regular c/osed iff F = Cl(Int(F))

Lemma 2.6. Lel (X, r) be a space with F 드 X. (1) N( 서 F) 드 N(r)JF (2) [f F is regular c/osed, th en N(rJF) = N(r)JF. Proof (1) It is well known and not difficult to show that a subset N of COIltamnollemptya space (Yopen, σ) IS setnowhere whose denseIntersection lfr every wlth nonempty N Is empty- open setNow let E E 36 T.R. Hamlett and David Rose

N(rlF) and let U E r - {0}. 1f U n F ￥ 0, there exists V E r such that O￥ VnF 드 U n F and (V n F) n E = 0 • V n E = 0. Let W = U n v, then W 드 U, W E r - {0} since W n F = V n F # 0, and W n E = ø since W 드 V. If U n F = 0, then U n E = 0. 1n either case, we conclude that E E N(r) and hence E E N(r )IF. (2 ) We need only show N(r)IF 드 N(rIF). Assume E 드 F and E f/. N(rIF); i.e. , IntF(CIF(E)) ￥ 0. Now let IntdCl(E)) = W n F where W E r. Since F is regular closed, we must have W n Int(F) ￥ m and hence ø￥ wnlnt(F) 드 wnF = IntF(Cl(E)) 드 CI(E) • O￥ Wn

Int(F) 드 Int(Cl(E)). Therefore E f/. N(r)IF and N(r)IF 드 N(rIF) ‘ If (X,r) is a space with F a closed subset of X , then (F,r lF) is QHC iff (F, r lF) is N(rlF)-compact by Theorem 1.1. T hu s (F, rlF) being QHC • F is N(r)IF-compact (since N(rlF) 드 N(r)IF) in (F,r lF) and hence F is N(T)-compact as a subset of (X,r ). 1n particu lar, H-closed subspaces are N(r)-compact as subsets. Thus 10 α lly H-closed spaces are locally N (r)-compact. An example of an N(r)-compact subset whi ch is

not an H-cl osed subspace is provided in [3]‘ However, we can prove the following result

Theorem 2.7. Let (X,T) be a Hat때 orff space. Then (X, r) is locally H-closed 얘 (X, r ) is locally N (r)-compact Proof 1n light of the above remarks, we need only show sufficiency. As­ sume (X, r ) is locally N( r )-compact and let x E X . By Theorem 2.5, there exists U E r (x) such that Cl*(U) is N(r)-compact. Let F = Cl‘ (U). By Theorem 2.3 , (F,rIF) is N(r)IF-com pact as a subspace. H follow s from Theorem 9 of [14] that CZ*(U) = CI(U) , hence F is regular closed. By Lemma 2.6, N (r)I F = N(rIF). Thus (F,rIF) is N(rlF)-compact as a subspace and hence H-closed as a subspace by Theorem 1.1. 1n Example 3.4 of [12], Porter exhibits a closed subspace F of an H­ closed space whi ch is not locally H-closedj in particular, the subspace is not H -closed. It is shown in [3] that a r* -closed subset of an I -compact space is I-compact. Thus the closed suhspace of Portel'’ s example is N( r )­ compact and not N(rlF)-compact, hence N(rIF) # N(r)IF in this exam­ ple. This shows that the regular closed hypothesis is necessary in Lemma 2.6, (2) Local compactness with respect to an ideal 37

3. Subspaces It is well known that a subspace of a Hausdorff space is locally compact implies it is locally closed (the intersection of an open and closed set), and in a locally compact Hausdorff space, the intersection of an open set with a closed set is locally compact. Porter [12] shows that in a Hausdorff space, a locally H-closed subspace is locally closed and the converse doesn’t hold even in an H-closed space. More generally, we have the following.

Theorem 3.1. Lel (X,r,I) be a Hausdorffspace. Th en A 드 X is locally I -compact implies A = V n F wh ere V E r and F is r* -c/osed Pmof It suflìces to show that A is rI CI*(A) -open. Let x E A, then there exists U E (rIA)(x) such thal CIÄ( U) is I-compact. Note that (rI A)"(IIA) = r‘ IA. Let U = A n V where V E T. Then CI*(A n V) n A = CI*(U) n A = C따 (U) and hence CI’ (A n V) n A is r*-closed. Now AnV 드 CI*(A n V) n A 드 A • CI‘ (A n V) 드 CI*(A n V) n A 드 A • CI*(A) n V 드 A since CI‘ (A ) n V 드 CI*(A n V) . Thus CI‘ (A) n V is a rI C I" (A) nbd of x conlained in A, showing that A is r ICI" (A)-open‘ An immediate consequence of Theorem 3.1 is the observation that a r*-dense lo cally I -compact subspace of a Hausdorff space must be r-opcn

The converse of Theorem 3.1 is fal se as the subspace X - 5 of Example 1 shows.

Example 2. Let X , O", I , r , and 5 be as in Example 1. X - 5 is an open subspace of (X, r) which is I-compact and T2 , but 0 E X 一 5 has no II(X - 5)-compact nbd. Indeed, letting V(a) = [0, a) - S we have that {V(a): a E S} is a rl(X -S)-nbd base at 0 and Clx_s(V(a)) = [O ,a) - 5 is not I I(X - 5)-compact (note that I I(X - 5) = {0}) for any a E S . Thus there does not exist a rl(X - S) nbd U of 0 such that Clx_s(U) is II(X - 5)-co mpact, and hence the open subspace (X - 5끼 (X - 5)) is not locall y II(X - 5)-compact. Lemma 3.2. Let (X , r ) be a space. If U E r then N(r )IU = N (r lU). Pmof We need only show N(r)1U ç N(rIU). Let A 드 U. Then Intu(C1u(A )) = 1 η tu[Cl(A) n U] = Int[CI(A) n U] 드 I 떠 (CI(A) ) . Thus if A E N(r) IU , then A E N(rIU) Example 3.4 of [12] provides an example of an open subspace U of an

N(r)-compact T2 (actually an H-closed sem ireg 비 ar) space which is not 38 T.R. Hamlett and David Rose locally H-closed. This subspace cannot be locall y N(r)-compact. For i[ it were, each point x in the subspace would have a regular closed JV(r)IU compact nbd F. By Lernma 3.2, F would be N(rIU)-compact. Now N(rIU)IF = N((rIU)IF) since F is regular closed • F is an H -closed subspace o[ (U, rlU) and U would be locally H- closed. This contradiction sh 。、，vs that U is not locally N( r )-compact. A partial converse to Theorem 3.1 is provided by the following three theorems.

Theorem 3.3. Let J( be an I-compact subset 01 a space (X ,r ,I). 11 A is r* -closed, then J( n A is IIA-compact in (A , rIA). Proof Let {U" n A : (} E <"'>} be an open cover o[ f( n A in (A , rIA). Then {U,, : (} E <"'>} U{X -A} is ar‘ open cover o[ J( and hence there exists a finite s 배 co ll ect i on {U,,;: i = 1,2,'" ,n}U {X -A} which covers all o[ J( except for a set in I. The fin 따 subcollection {U,,;nA: 1,2, ... ,n} must then cover all o[ J( n A except for a set in I

Theorem 3 .4. Let (X, r , I) be locallν I -compact with A a r*-closed subset 01 X . Then (A , r IA,IIA) is locally IIA- cor깨 act. Proof Let x E A. There exists U E r(x) and an I-compact subset F o[ X such that x E U ç F. Now x E U n A 드 F n A, U n A E (r IA)(x) and F n A is IIA-compact by Theorem 3.3. Given a space (X, r) and letting N(r) denote the ideal of nowhere dense sets, the topology r* (N(r)) is known in the literature as 꺼 and the r* -closed sets are called (}- c/osed

Corollary 3.5. Let (X,r) be a locally H- c/osed Hausdorff space ω ith A an Q-c/osed subset 01 X. Then (A,rIA) is locally N(r)IA-compact. Proof This [ollows immediately from Theorem 2.7 and Theorem 3.4. Note that in Example 2, r is not regular with respect to r*.

Theorem 3.5. Let (X , r,I ) be locally I-compact with r reguiar ψ ith res pect to r ‘ . JI A E r , then A is locallν IIA-compact. Proof Let x E A, then there exists U E r(x) and a r*-closed subset F such that x E U 드 F 드 A. Sin ce (X, r) is locally I-compact, there exists V E r(x) and an I-compact subset J( such that x E V 드 f(. Nowwehave xE(UnV) 드 (I( n F) 드 F 드 A, J( n F is IIF-compact in F (Theorem 3.3), and hence f( n F is IIA-compact in A (Theorem 2.3). Thus J( n F is an IIA-compact nbd of x in (A , rIA). Loca l compactn 앙 5 with respect to an ideal 39

4. Preservation by Functions

It is well knα，v n that local compactness is preserved by continuous open surjections. In this section we prove an analagous result in a more general setting. Given spaces (X,T ,I ) and (Y,O'), a function f : (X,T ,I ) • (Y, a) is said to be pointwise I-continuous [5] (abbreviated PIC) iff f : (X, T*) • (Y, a) is con t inuo 야 . It is easily verified [9] that f(I) = {f(I) : 1 E I} is an ideal on Y. We will say the function f is pointwíse I -open (abbreviated PIO) iff f: (X,T) • (Y, a* (f(I))) is open. We will state the following easily verified lemma without proof

Lemma 4.1. Let (X,T,I) be a space. Jf A 드 X , then A is an I -compact subset Of(X,T) iff A is an I-compact subset Of(X,T“) . Lemma 4.2. A space (X, T,I) is locally I -compact íff(X, T* ,I) is locally I-compact. Proof Necessity. Assume (X,T,I) is locally I-compact and let X E X Then there exists U E T(X) and an I -compact, in (X,T) ,I( 드 X such that X E U 드 K. By Lemma 4.1 , K is I-compact in (X,T*) and hence (X, T*) is locally I -compact Sμ:fficiency. Assume (X, T* , I) is locally I -compact and let X E X Then there exists U E T, J E I , and an I-compact, in (X, T*) , K 드 X such that X E U - J 드 K. Now x E U 드 K U l , K U J is I -compact in (X , T) and the proof is complete.

Theorem 4.3. Let f : (X, T, I) • (Y,a) be a PIC fuπ clion from the space (X,T,I ) to the space (Y,a). Jf A 드 X is I-compact , then f(A) 드 Y is f(I)-compact

Proof Let {U" : a E l:>} be a a-open cover of f(A) . Then U- 1 ( κ) : a E l:>} is a T* -open cover of A. By Lemma 4.1 , there exists a finite su bcollection {f-l (V"‘) : i = 1,2," . , n} such that A - U;j-l (V".) E I . Now, since f (A) - U;V"‘ 드 f [A - U;j-' (V".)] E f (I ), we ∞ nclude that f(A) is f(I)-compact by heredity of f(I).

Theorem 4 .4. Let f (X, T,I) • (Y,O',f(I )) be a PIC and PIO surjection. lf (X, T,I) is locallν I -compact, then (Y, a,f(I)) is locally f(I)- compact. Proof Let y E Y and let x E f-l(y). There exists U E T(X) , K an I­ compact subset of X , such that X E U 드 K; hence, f(x) E f(U) 드 f(K). 40 T .R. Hamlett and David Rose

By.y Thu T hesor (eYm, a* 4).3 i s alndoc allLye mmaf (I ) -4co.1m, pactf (K )a nisd anhen fce(I ()Y-com, a ) pisact locall a* -y뼈 f (I )­ compact by Lemma 4.2. The classical result that local ∞ mpactness is preserved by continuous open surjections is obtained from Theorem 4.4 by letting I = {0}. Given a space (X ,T), we noted earlier that T* (N(T) ) is known in the literature as 까 Also, a fun ction f (X,T) • (Y,a) is known as an o: -homeomorphism iff f (X, T" ) • (Y, a") is a homeomorphism It is shwon in [4] that a function is an o: -hom eomor p hismκ it is a seπmT따l homeomor아찌r디대phi s m s (o: -hom∞ morphi s ms ) are known as semi-lo1Jological pmp­ erties [1] (o:-topological properties)

Corollary 4.5. Local nowh e π dense compactness is a semi-topological φ - topological) and hence topological properly. Proof Let f : (X, T) • (κ 。) be a semi-homeomorphism. Assume that (X , T) is locally N(T)-compact. A semi -homeomorphism (o:-homeomorphism) is both PN(T) C a.nd PN(T)O, hence by Theorem 4.4 (Y,a) is 10ca.lIy f(N(T))-compact. By Theorem 1.2 of [1] , f(N(T) ) 드 N (a) , thus we ha.ve (Y, a) is locally N( a )-compact

Given topological spaces (X, T ,I) and (Y, σ) ， a bijection f : (X, T , I) • (Y,a) is called a "-homeomorphism with respect to T, I , and a [4] if f : (X, T*(I) ) • (κ 。"(I (I))) is a homeomorphism. Properties preserved by "-homeomorphisms are ca.lIed *-topological properties [4 ].

Corollary 4.6. Local I-compactness is a "-lopological pmpe7"i y. Pmof Note that • -homeomorphisms are both PIO and PIC and apply Theorem 4.4

Corollary 4.7. Jn lh e catego 깨 of Hausdorff spaces, /ocal H -closedness is a semi-topological(o: -top%gical) properly.

P7"O of Corollary 4.5 and Theorem 2.7.

5. Products

Given a family of spaces {(X", T,,) 0: E 6 }, it is well known that mX a, n T,,) is 10cally compact (i.e. 10cally {0}-compa.ct) iff each (Xa, Ta ) is locally compa.ct and all but fìnitely many X" are compact. Obreanu [11] proved tha.t a product of Hausdorff spaces is locally H-closed (10- Local compactness with respect to an ideal 41

cally nowhere dense-compad) iII each factor is locall y H-closed (locally nowhere dense-compact) and all but finitely many of the factors are H closed(nowhere dense-cor매 act) ‘ In our more general settin g, we have the íollowing resul ts

Theorem 5.1. Let {(X""r",) ‘ o E l',.} be a co /leclion of topological spaces If (IT X"" Il r"" T ) is 10ca/lyT-compact, then (1) each (X"" r"" P,,(T )) is locally P"， ( T)-COl때acl ψ here P" is the pro­ jection map onto X ,,; aηd

(2) all but finitely ma째 X ", are P,,(T )-compact

Proof Let x E Il X ", and let W be an T-compact nbd of x . vV contai ns a basic nbd of the form

P;;,l(U",) n p，김 (Ua2) n n P꾀 (U"，.)

and if 0 rt {o" 02, ... , On} then P,, (W) = X ",. For each 0 rt {01 , 02, . . . ,On },

X ", is P,, (T)-compad by Theorem 4.3; and for each o;, X"" is locally P",, (T)-compad by Theorem 4.4. The classical res u1t for local compactness is an immedi ate co rollary obtained by taking T = {0} . The Obreanu res ult [11] cannot be obtained as an immedi ate corollary since the projection maps do not necessarily preserve nowhere densene8s.

Theorem 5.2. Let {(X"" r"" T,,) : 0 E l',.} be a colleclion of spaces ψhich are locally T", -compact for each 0 , and for all 0 rt s = {01, 02, . . . , On} let X" be T", -compacl. lf T is an ideal on Il X ", such that P;l(T",) 드 T for every 0 , then (Il X "" Il r""T) is locallν T-compact. Proof Let x = (x", ) E Il X ", . For each x"', there exists an T", -compact sub8et f{"" of X "" and U" , E r",(x" ,) 8uch that x", E U" , 드 f{" ,. Now

letting f{ = f{"" x !( "'2 x ‘ X f{"," x Il {X", : 0 rt s }, we have that J( i8 T- compact as a su bspace of Il X", by the generalized TychonolI Theorem

of Rancin [13], and hence is an T-compact nbd of x in Il X ", . The classical result concern ing local compactness is obtained as an immediate corollary by lettin g each T", = {0} and T = {0} . To obtain the res ult of Obreanu [11] concern ing locally /f-c1osed spaces we nced the following lemma whose ea8y proof is omitted

Lemma 5.3. If f : (X,r) • (Y,O") is continuous and open and B E N(O"), then f -1(B) E N(r). 42 T.R ‘ Hamlett and David Rose

Corollary 5 .4[11]. Let {(X",T,,): a E 6} be a collection of (Hausdorff) locally H -c/osed spaces all but a finit e number of ψhich are H - c/osed. Then mX" , n T ,,) is locally H -c/osed. Proof The result follows immediately from Theorem 2.7, Theorem 5.2, and Lemma 5.3.

Acknowledgement ‘ The authors wish to thank Prof. Dragan Jankovi c’ for hel pful suggestions and comments.

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