© Hindawi Publishing Corp. LOCAL COMPACTNESS IN APPROACH SPACES II

This paper studies the stability properties of the concepts of local compactness introduced by the authors in 1998. We show that all of these concepts are stable for contractive, expansive images and for products. 1. Introduction. Having introduced notions of local compactness, basis local compactness and related measures, and having studied the basic relationship among these concepts in [3], in this paper we study stability properties. Keeping in mind the stability properties of local compactness in TOP, the category of topological spaces and continuous maps, we study how our notions behave under mappings and products. Especially, in the topological case, the maps which are required in order to preserve local compactness are continuous open surjections. This necessitated first finding out what is the right notion of open map in AP. This is duly done in this paper, and it turns out that the notion which we use has nice characterizations for a number of primitive approach structures. Of course, it also turns out to be the right concept in connection with local compactness.


INTRODUCTION
The purpose of this paper is to introduce and study some reasonable definitions of a concept of local compactness in approach spaces.The search for the right notion if such a uniquely determined generalization exists is motivated, not only by the obvious fact that local compactness is an important and natural concept in topology and hence, as has been made clear by the development of the theory so far, will be equally important in approach spaces, but more specifically it is motivated by the search for a description of the exponential objects in AP.In [1] this problem was successfully solved in PRAP, the category of pre-approach spaces.As is well-known however the situation in TOP as compared to PRTOP, the category of pretopological spaces, is considerably more complicated, and the same pattern presents itself in the theory of approach spaces.The link between notions of local compactness and exponential object in AP will be the topic of forthcoming work.Unrelated to this problem however, we found that there are a number of intuitively appealing concepts in AP which have nice properties and which even allow for quantification in the way Kuratowski's and Hausdorff's measures of non-compactness quantify the topological notion of compactness.In this first part of our paper we concentrate on a basic study of local compactness concepts and their relationship.

PRELIMINARIES
Given a set X we denote its power set by 2 x and the set of its finite subsets by 2 (x).We recall those concepts and results from Lowen [2, 3] which we require in the sequel.A map :X x 2 x [0, c] is called a distance if it fulfils (D1) VA 2X,Vx X x A 5(x,A) O.
If is a distance on X and A C X, the function 6A X -[0, oo] is defined by 5A(x) := 5(x,A).
(AS) Vx X, Vo A(x), VN 10, 3(v,),ex I] A(z), Vz, y X: o(y) A N < x(z) + o,(y).zEx The elements of an approach system are called local distances.For ease in notation we shall, whenever convenient denote an approach system (A(x)),ex also simply A. If A is an approach system then A := (A(x))ex is called a basis for A if it fulfils the properties (bl) Vx X A(x) is a basis for an ideal.
$.If 6 is a distance on X then the system At where for all x X A,(x) { [0, o01 x VA c X-inf o(a) < 5(x,A)} aA is an approach system on X.
A set X equipped with an approach system or equivalently a distance is called an approach space and is usually denoted (X, Jl).The associated distance is usually denoted simply 6 instead of 6A and analogously if 5 is the primary defined structure, A is usually simply denoted ,4, unless confusion might occur.
PROPOSITION 2.2 [3] If A is a basis for ,4 then .is also obtained by (,a) sup If (X, ,4) and (X', 4') are approach spaces and f X X' is a map then f is called a contraction if it fulfils any of the following equivalent (see Lowen [3]) conditions: (el) Vz X, Vo' A'(f(x)) o' o f (c2) For any basis h' for Ar, Vx X, re' A'(f(x)) ' o f e A(x).
Approach spaces and contractions form a topological construct Lowen [3] which we denote AP.For categorical concepts, in particular topological categories, we refer to Admek et al. [4].TOP can be embedded as a bireflective and bicoreflective subconstruct of A P. The embeddingsfunctor is given by (X, 7") --+ (X, AT) leaving morphisms unaltered and where AT is the approach system AT(x) := {o [0, c] x o(x) 0, o u.s.c, at x} for every x X, and which has as basis the collection {Or V neighborhood of x for T} for every x X, where Or(x) 0 if x V and Oy(x) cx if x V.The associated distance is given by 6(x, A) 0 if x and 6(x, A) if x .T he subconstruct thus obtained is isomorphic to TOP.We recall that the bicoreflection of (X,A) in TOP is given by idx (X,,4-a) -+ (X, JI) where 7 is the topology on X determined by the neighborhood system or equivalently by the closure operator := {x X (x, A) 0} for every A 2x.
The construct pq-MET of extended pq-metric spaces and non-expansive maps too can be embedded as a bicoreflective subconstruct of AP.The embedding is given by (X, d) -- (X,)   leaving morphisms unaltered and where Ad is the approach system J[d(x with obvious basis consisting of the single element d(x, .).As to be expected the associated distance is given by 6(x, A) infaea d(x, a).
As for topological spaces a convergence theory can be developed in AP (see E. and R. Lowen [5, 6] for more details).The difference with topological spaces however is that with each filter and each point we can give a distance the point "is away from being a limit point" of the filter.Precisely this goes as follows.Given a set X, F(X) is the set of all filters on X; if " E F(X), then U(') is the set of all ultrafilters finer than '.If C 2 x then stackx := {B C X G E G c B}, if G consists of a single set G we write stackxG and if moreover G consists of a single point a, we write stackxa for short.If no confusion can occur, we drop the subscript X.Also, if F C X we abbreviate U(stack F) by U(F).The set sec " is defined as the union of all ultrafilters finer than ', which means sec " := {A C X VF E " A A F # 0}.Let (X, 5) be an approach space and :F F(X), then the limit (-function) of is defined as '(z) := sup (f(z, A), Vz X.

AEsec
It will be useful also to have a description of 6 not in terms of 6 but in terms of the associated approach system t.
PROPOSITION 2.3 [7] Let (X, .A) be an approach space.For any yz E F(X) and x E X we have AAgV(x)= sup inf_supo(y) where both .4 and .T may be replaced by bases.
It is worthwile to mention that limits also as in TOP provide yet a third way to describe approach spaces (see E. and R. Lowen [5] for more details).For us it suffices to mention that the distance 5 can be recovered from 6 by (x,A)= inf )lJ(x).

uu(A)
Convergence in the topological bicoreflection (X, 7) of an approach space (X, 5) can easily be derived from the limit associated with 5.If " F(X) then "x in T if and only if (x) 0.
In the case of cx)-pq-metric spaces the associated limit takes on a more simple and intuitive form.If " E F(X) then Y(x) inf sup d(x, y), F. yF and in case " is generated by a sequence (x,), this simply means that '(x) lim sup d(x, DEFINITION 2.4 Given an approach space (X,A), we define the measure o.f compactness oI X as #c(X) sup inf /(x).

uev(x) ex
The idea behind this definition is the following.Compactness means every ulrafilter should have a convergence point.Therefore the information given by #e is based on the verification for all ultrafilters what are their "bes convergence" points.Before all else we give a number of equivalent forms of this definition.
PROPOSITION 2.5 [2] For any approach space (X, riO, we have (X) where A is a basis for the approach system A. 1.If (X, AT) is a topological approach space then (X, T) is compact if and only if u(X) O.
2. If (X, Aj)Ij is a family of approach spaces then

SOME NOTIONS OF LOCAL COMPACTNESS IN AP
In this section we will define some notions of local compactness and basis-local compactness in AP, which on topological spaces coincide with the topological notions of local compactness and basis-local compactness.We will denote the notions of local compactness by LCn where n is a number between 1 and 5, and the associated notions of basis-local compactness likewise by BLCn.DEFINITION 3.1 Let (X,,4) be an approach space.
1. (X, ,4) is LC1 if and only if its topological coreflection is locally compact.
5. (X, ,4) is LC5 if and only if VxX, Ve>0" inf pc(F)< inf It is easily verified that the given definitions of LC2 are equivalent, and likewise are those of LC3.
We have the following obvious relations between the different LCn.
PROPOSITION 3.3 Let (X, T) be a topological space.Then (X, T) is locally compact (X, AT-) is LC (X, AT-) is LC5.This is a straightforward result and it shows the LCn can be considered as generalizations of topological local compactness in the context of approach spaces.Notice that for a topological space, Ve(x) V(x) for every e > 0 and every x X. PROPOSITION 3.4 1.Let (X, d) be an oc-pq-metric space.Then (X, d) is LC2 if and only if every point possesses a totally bounded neighborhood.

GCF
If an approh space is BLCn, then it is also LCn.In order to see this, notice that we have Ye(x) ,<e,(x) for eve x X and > 0. The relations between the different BLCn are for the LCn: PROPOSITION 3.7 Let (X, T) be a topologil sce.
This rult illustrates the BLCn can be ewed generalizations of topological bis-local com- ptness in the context of approach spaces.For --metric spaces, we get the following result.
PROPOSITION 3.8 1.Let (X, d) be an -pq-metric space.Then (X, d) and only i/every point possesses a neighborhood b coisting of totally bounded neighbor- hoo&.
For some notions of local compactness in AP we can introduce a meure of local comptns which is a generalization in which all the nice properti are preserved.DEFINITION 3.9 Let (X, A) be an approach space.
The following inualities can eily be check.
PROPOSITION 3.11 Let (X,A) be an appwach space.Then B In E. and R. Lowen [6] it was shown that (X, A) is an approach space.Moreover, if H is an ultrafilter on X and x 6 X, f(x) if'CHandH#stackx AH(x) o if " /2 and H # stack x 0 if H stack x.If there exists an x 6 X such that H stackx, then inf,x A/g(x) 0. If for every element x e X, H # stack x and Y C , then inf, ex AH(x) infzex f(x).If for every element x of X, /4 # stack x and " fZ H, then infzx A/4(x) oo.So we get the following three cases.If X is finite (i.e., all ultrafilters on X are point filters), then #c(X) supuev(x) inf, ex AH(X) 0. If X is infinite and if for every ultratilter H on X, not being a point filter, " C H, then p(X) influx f(x).Finally, if X is infinite and there exists some ultrafilter/4 on X, not a point filter, such that " if H, then #(X) oo.Let B be the intersection of all ultrafilters on X which are not a point filter.A straightforward verification shows that B {X \ A A is finite}.Using this fact, we can state 0 if X is finite #c(X) inf,x f(x) if X is infinite and Y C {X \ A A is finite} c if X is infinite and 9 v {X \ A A is finite}.