Topological subspaces of free topological groups
by
Kohzo Yamada
Shizuoka University

Let F(X) and A(X) be respectively the free topological group and the free abelian topological group on a Tychonoff space X. In this talk, we discuss the following topological properties:

(1) locally compactness, (2) Cech-completeness, (3) point-countable type, (4) first countability, (5) Fréchetness, (6) sequentiality, (7) k-property

However there are some concrete spaces X such that F(X) and A(X) have the properties (6) and (7), it is known that neither F(X) nor A(X) on a non-discrete space X has the properties (1) ~ (5). On the other hand, if X is a compact metric space, then both of F_n(X) and A_n(X) have these properties for each n in \omega, where F_n(X) and A_n(X) are subsets of F(X) and A(X), respectively, formed by all words whose reduced length is less than or equal to n. In this talk, we show that there is a subspace of F(X) (A(X)) which is not contained in any F_n(X) (A_n(X)) such that it has the properties (1) ~ (5), every such a subspace with the properties (1) ~ (4) must have a special form in F(X) (A(X)) and there is a Fréchet subspace of F(X) (A(X)) without the form such that it isn't contained any F_n(X) (A_n(X)).

Recently we obtained the equivalent conditions of a metrizable space X such that every or some F_n(X) (A_n(X)) is metrizable and proved that for a metrizable space X and n >= 2, if F_n(X) (A_n(X)) is first countable, then it is metrizable, respectively. Here, we also discuss the conditions of a metrizable space X such that every or some F_n(X) (A_n(X)) is Fréchet.

Date received: June 28, 1999