Metrizability of GO-spaces and topological groups
by
Yoshio Tanaka
Department of Mathematics, Tokyo Gakugei University, Tokyo
Coauthors: Chuan Liu (Ohio University), Masami Sakai (Kanagawa University)

We give metrizability theorems on GO-spaces or topological groups in terms of weak topology.

Definition: Let P be a cover of a space X. Then, P is a k-network for X, if whenever K subset U with K compact and U open in X, K subset \cup P' subset U for some finite P' subset P. As is well-known, CW-complexes, Lasnev spaces, or quotient s-images of metric spaces are k-spaces having a point-countable k-network.

Theorem: Let X be a GO-space. Then, (1), (2), and (3) below hold.

(1) Let X have a point-countable k-network. Then X is a paracompact space with a point-countable base if the following property (a), (b), or (c) holds. In particular, if X has a \sigma-compact-finite k-network, then X is metrizable.

(a) Each point of X is a G_\delta-set.

(b) X is a quasi-k-space.

(c) t(X) <= \omega.

(2) Let X have a point-countable k-network. Then X is metrizable if the following property (d), (e), or (f) holds.

(d) X is a (locally) separable space.

(e) X is a (locally) w\Delta-space.

(f) X is a (locally) \Sigma-space.

(3) Let X be a topological group. Then X is metrizable if one of the above properties (a) ~ (f) holds.

Date received: June 21, 1999