Spaces in which Point-finite Open Covers have Finite Subcovers
by
Nurettin Ergun
Istanbul University
Coauthors: Stephen Watson

G. Aquaro has proved in 1965 that point-countable and in particularly point-finite open covers of a countably compact space have finite subcoverings. Actually a countably compact space X without any separation axiom satisfies the following covering property: For any open covering U of X there exists a finite set F of X such that X = st(F, U) = \cup {st(x, U) : x in F}. The last mentioned covering property determines a new and strictly larger class of starcompact spaces which is defined by Fleischman [Fl]. He proved that there exists a starcompact non countably compact Hausdorff space. Another related well known result has been proved by Iseki and Kasahara in 1957: A regular Hausdorff space X is countably compact if and only if every point-finite open cover of X has a finite subcover. Z. Frolik proved on the other hand in 1960 that this equivalency does not necessarily hold without regularity condition. He defined a non countably compact Hausdorff space in which point-finite open covers have finite subcovers. Actually point finite open covers of this space are necessarily finite. One must notice in here that his non regular counterexample space is not even \aleph₀-collectionwise Hausdorff, see [Fr] or [E] page 241. As is well known a topological space X is called \aleph₀-collectionwise Hausdorff if points of any countable closed-discrete subset is separated by suitable pairwise disjoint nbhds of these points and it is also well known that regular Hausdorff spaces are \aleph₀-collectionwise Hausdorff. On the other hand there exist \aleph₀-collectionwise Hausdorff (even collectionwise Hausdorff) non-regular Hausdorff spaces. In fact any countably compact Hausdorff space is collectionwise Hausdorff since closed-discrete subsets are necessarily finite in such spaces, but, examples of non-regular countably compact Hausdorff spaces are already known. Take for instance X = {p} \cup (w₁ ×[0, 1]) where p \not in w₁ ×[0, 1] and define the basic nbhd's of the point p as G_\alpha(p) = {p} \cup ([\alpha+1, w₁) ×[0, 1)) for each \alpha < w₁ and let w₁ ×[0, 1] be equipped with the usual product topology. This countably compact Hausdorff space is not regular since p and w₁ ×{1} are not separated in X.

In this note we first give a generalization of Iseki & Kasahara theorem. We also prove the existence of a Hausdorff space X in which 1) point-w_\alpha open covers have a subcover with cardinality less that w_\alpha, 2) X has a closed-discrete subset with cardinality w_\alpha and finally 3) X has a point-w_\alpha open cover with cardinality >= w_\alpha. It is also shown that any Hausdorff space X can be embedded as a closed nowhere dense subset of a Hausdorff space Y_X such that Y_X-X=I is a set of isolated points and any open family of Y_X covering X and point-finite (resp. point-countable) on I has a finite (resp. countable) subfamily covering X. Additionally we proved that any countably compact Hausdorff space is a closed nowhere dense subspace of a Hausdorff space in which point countable open covers have finite subcovers. We finally posed two related conjectures.

Date received: April 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-05.