Frechet-Urysohn for finite sets
by
Paul Szeptycki
York University
Coauthors: Gary Gruenhage

For a space X and a point x in X, a family P of subsets of X is said to be a pi-network at x if for each open U containing x, there is p in P such that p subset U. We will say that a family P of subsets of X converges to x if for each open U containing x, {p in P:p Ë U} is finite. If P consists of singleton sets, then P converges to x if the sequence formed by any enumeration of the singletons converges to x.

Reznichenko and Sipacheva defined a space X to be Fréchet-Urysohn for finite sets (FUF for short) if for each x in X and each P subset [X] ^{< \omega}, if P forms a pi-network at x, then P contains a subfamily that converges to x.

This strengthening of the Frechet-Urysohn property is related to at least three topics: metrizability of topological groups, productivity of Frechet-Urysohn property, and the point-open games G_{O, P}(X, x). We will explain these relationships and emphasize a number of open problems.

Date received: February 26, 2003

Copyright © 2003 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 # cajz-70.