Connectifications of Metrizable Spaces
by
Gary Gruenhage
Auburn University
Coauthors: Attilio Le Donne

A space Y is called a connectification of a space X if X is dense in Y and Y is connected. It is easy to see that if X has a compact open subset, then X has no Hausdorff connectification. A number of recent papers have studied the problem of when a space X with no compact open subsets has an Hausdorff (at least) connectification. Here we consider the natural question whether every metrizable space with no compact open sets has a metrizable connectification. This question (and also one asking if there is at least a Tychonoff connectification) was stated in a paper of Alas, Tkachuk, Tkacenko, and Wilson, who proved that the answer is positive for separable metric spaces. Also, Porter and Woods have shown that every metrizable space with no compact open subsets has a Hausdorff connectification, and Neumann-Lara and Wilson have shown that every strongly 0-dimensional metrizable space has a Tychonoff (orderable) connectification.

Here we show that a metrizable space X has a metrizable connectification if (and only if) X can be embedded in a metrizable space Y such that every clopen subset of X has a limit point in Y. An easy corollary is that every nowhere locally compact metrizable space has a metrizable connectification. The result for separable metric spaces is another easy corollary, and provides a quite different method of proof from the one given by Alas et al.

Date received: April 12, 1996

Copyright © 1996 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 # caae-27.