First countable extensions of regular spaces
by
Petr Simon
Charles University
Coauthors: Gino Tironi

A space X is feebly compact if every locally finite collection of nonempty open sets is finite. A space X is pseudocompact if it is completely regular and every real-valued continuous function defined on X is bounded. An extension of a space X is a space Y containing (a homeomorphic copy of) the space X as a dense subspace.

We will present a technique, which allows to find a feebly compact, (pseudocompact, resp.) envelope of a given space, which preserves first countability or even Mooreness of the input space. Our technique allows for

Every locally feebly compact regular space X can be embedded as a dense open subspace in a feebly compact regular space Y which is first countable at every point from Y\X.

Every separable, locally feebly compact Moore space embeds as an open dense set in a feebly compact Moore space.

Every locally pseudocompact, separable Moore space embeds as an open dense set in a pseudocompact Moore space.

There is a Moore connected pseudocompact space without a dense relatively countably compact subset.

The first two theorems answer Stephenson's questions 23 and 25 from [S] and significantly strengthen [T, Theorem 2.2]. The third theorem answers Reed's questions 4.(7) from [R2] and 5.2 from [R3]. The last theorem improves [W, Theorem 3.1] by removing the assumption of CH.

[R1] George M. Reed, On chain conditions in Moore spaces, Gen. Top. and its Appl., 4(1974), 255 - 267

[R2] George M. Reed, On subspaces of separable first countable T₂-spaces, Fund. Math., 91(1976), 189 - 202

[R3] George M. Reed, Set-theoretic problems in Moore spaces, Open Problems in Topology, ed. by J. van Mill and G. M. Reed, North-Holland 1990, 163 - 181

[S] Robert M. Stephenson, Jr., Moore-closed and first countable feebly compact extension spaces, Topology Appl., 27(1987), 11 - 28

[T] Ian J. Tree, Extending the discrete finite chain condition, Topology Appl., 52(1993), 267 - 278

[W] W. Stephen Watson, A connected pseudocompact space, Topology Appl., 57(1994), 151 - 162

Date received: May 19, 2003