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.