ON WEAKER FORMS OF COMPACTNESS LINDELFNESS AND COUNTABLE COMPACTNESS

A theory of e-countable compactness and e-Lindelfness which are weaker than the concepts of countable compactness and Lindelofness respectively is developed. Amongst other results we show that an e-countably compact space is pseudocompact, and an example of a space which is pseudocompact but not e-countably compact with respect to any dense set is presented. We also show that every eLindelof metric space is separable.


INTRODUCTION.
Using the terminology and notation in [3], it is known that a topological space X is compact iff its enlargement *X contains only near-standard points, and that a subset A of a regular Hausdorff space is relatively compact Iff *A contains only near standard points.Hechler [I] wanted to know what this condition implied topologically in not necessarily regular spaces.
He was led to the notion of what he called 'ecompactness' which is weaker than the notion of compactness.It is the purpose of this paper to develop a theory on the analogous concepts of e-countable compactness and e-Lindelofness in the spirit of Hechler's study of e-compactness [I].
In particular we extend the well-known result that a countably compact space is pseudocompact to an e-countably compact space is pseudocompact.We also show that the Lindelof condition in the theorem that every Lindelof metric space is separable can be weakened to e-Lindelof.
Let D be a dense subspace of a topological space X.
DEFINITION 2.1. ([|I): (a) X is said to be e-compact with respect to 0 if each open cover of X contains a finite subcol[ect[on that covers D. (b) X is e-regular with respect to a dense subset D if for each closed FoX and each p F there eist disjoint open sets U and V such that p U and F 0Dc V.By analogy we introduce the following: DEFINITION 2.2.
X is e-countably compact with respect to D if every countable open cover of X has a finite subcollectlon coerl, D.
DEFINITION 2.3.X is e-Lindelf with respect to D if every open cover of X ha n countable subcollectlon covering D. Recall Hechler's extension of the topology T for X: Let be a family of subsets of X, and let T(E)= {U-FIU,T and F Is a subset of a flnie union of members of }.By K() we shall mean K with th, extended topology T().Hechler used the construction of the extended topology T() to provide examples of spaces which are e-compact but not compact (see Ill p. 223.

RESULTS.
The following two theorems are analogous to the corresponding theorems on e- compactness (see [I]).
Let X be Lndelof.Then X() is e-Lindelof iff there exists a dense set D (with respect to X(E)) such that for every E g E, E O D is countable.
PROOF.Suppose X(E) is e-Llndelof with respect to a dense set D. Assume for some E g E, E D is uncountable.Then {X-(E-{x}) x E} is an open cover of X having no countable ubcover of D, contrary to E() being e-Lindelof Thus E O D is countable for all E g .
Conversely suppose END is countable for all E and D is dense with respect to X().
Suppose {U F A} is an open cover of X.
Then U ,m E A} covers X, and as X is Lindelof, there is a countable subcollectlon [U say, coverlng .
i i--I Now [Ui F! will cover all except at most countably many points {xj of O.
But {x.j }J-I c U, where is a countable subcollectlon of {Ua -Fa: a A} Thus showing that X(E) is e-Lindelf.
Let X be countably compact.Then X() is e-countably compact there exists a dense subset D (with respect to X()) such that for every set E g , Suppose X() is e-countably compact with respect to D. Conversely suppose E 0 D is finite for all E E E, and D a dense set with respect to X(E).Suppose {U i Fi}i= is a countable open cover of X(E).Then {Ui}i.covers X, and as X is countably compact there is a finite subcover {Ukl}ri=l of X.

Now {U k
Fki}r {xj }= covers all except at most finitely many points of D, say.
i t i=l But {xj}j c U where is a finite subcollection of {Ui Fi}t=I* Thus {Uki-Fkt}r U is a finite subcollectlon of {U i Fi} i=I i= covering D, showing X(E) is e-countably compact.
THEOREM 3.3.If X is e-countably compact, then X is pseudocompact.
PROOF.Suppose X is e-countably compact with respect to D. Lec f be a continuous of X.Every e-countably compact first countable Hausdorff space is e- regular (with respect to a dense set D).
PROOF.Let p F, F closed in X.Let B be a countable open neighbourhood base p.
Since X is Hausdorff, for each q E F there exists open G and B E B such q q that GqO Bq .Let {B_}qq F' which, being a subfamily of must be q is a countable family of open sets covering F so that {X-F}U {HBI B B'} covers K.
Since X is e-countably compact there exists a finite family "cB' such that Dc(X-F)U U{HB: B 8"}-Thus FNDcU{: B B'g V. Let N{B:B 8"} which is open and contains p.It is easily verified thatUNV .
We now show how separability relates to the generalizations of compactness and Lindelofness introduced above.
It is well known chat every Lindel'f metric space is separable.In fact the Lindel'f condilon can be weakened to e-Lindelofness as the following result shows.THEOREM 3.5.
If X is meCrlc and e-Llndel'f with respect co a dense set D then X is separable.

PROOF.
For each n E IN let U ={S(x, I/n) Ix E X}.Since U is an open cover of Then 3p and 3p + I. where PolS a prlme 3 Po-and Po o m om Hence Um,(m'+l) O U3(m'+l) is the required nelghbourhood of m'+l meeting no member of '.
This example is motivated by the result that a Llndelf countably compact space is compact.
The analogous statement that an e-Llndel'df, e-countably compact space is e- compact is not in general true as the following example shows: + Recall the Novak space (see [2] p. 134): Lec Z denote the positive integers with the discrete topology and S the Stone Cech compactiflcatlon + + of Z Let F be the family of all countably infinite subsets of S, well-ordered by the least ordinal r of cardinal 2 c card (S).
Let {PAIA F} be a collection of subsets of S such that card (PA) < 2c' PD C PA whenever D < A, and (PA 0 PA where is the unique extension to S of the continuous function f: Z + + + Z which permutes each odd integer n+l with its even successor, i.e. f(n) n + (-I) Then we define P U {PAIA F}, and then define Novak's space by XffiPU Z+.
Note that Cx(Z +) X ([2] p. 135), hence X is e-Lindel'f with respect Z + to Also X is countably compact ([2] p. 135), but as X is not compact, X cannot be absolutely closed (as a regular absolutely closed space is compact).Thus X is not e-compact with respect to any dense set.
Assume for some E g , E N D is infinite.Choose an lnfinlte sequence {x n IN#in EO D and n let F ix n e IN}.Then iX (F-x}): x F is a countable open cover of X with no finite subcover of D, contrary to X being e-countably compact.
By continuity of f, we have f(X) c [-n,n], showing f is bounded.It is well known that a countably compact first countable Hausdorff space is regular. n