Free topological groups and dimension

Abstract

ABSTRACT. For a completely regular space X we denote by F(X) and A(X) the free topological group of X and the free Abelian topological group of X, respectively, in the sense of Markov and Graev.Let X and Y be locally compact metric spaces with either A(X) topologically isomorphic to A(Y) or F(X) topologically isomorphic to F(Y). We show that in either case X and Y have the same weak inductive dimension. To prove these results we use a Fundamental Lemma which deals with the structure of the topology of F(X) and A(X). We give other results on the topology of F(X) and A(X) and on the position of X in F(X) and A(X).1. Introduction. The purpose of this paper is to continue the study of free topological groups begun by Markov [10] and Graev [6]. Graev showed that if X and Y are compact metric spaces such that A(X) and A(Y) are topologically isomorphic, then X and Y have the same dimension. In Theorem 2 we strengthen this result by showing that if X and Y are locally compact metric spaces such that A(X) and A(Y) are topologically isomorphic, then X and Y have the same weak inductive dimension. In Theorem 3 we give the analogous result

Local compactness in free topological groups

“…The next result from [1,8,7] provides important information about the topological structure of the subspaces F n (X) C F(X) and A n (X) C A(X) of elements of length precisely n. …”

“…exists a compact set C of countable character in X such that X' C C. Observe that P = (F 2 (X) \ Fi(X)) U {e} is a clopen subset of F 3 (X) (see [2,8]), and P is closed in F 2 (X). Hence F 3 (X) has countable type at the identity e and at each element of length 2.…”

Module-valued functors preserving the covering dimension

“…Since this class contains also the additive group R of reals with its standard topology, our result covers also the famous theorem of Pestov who accomplished the effort of a great number of mathematicians (see [1], [5], [9], [15], [16]) by proving that l-equivalence (that is R-equivalence in our notation) preserves the covering dimension. This result was later generalized by Gulko for u-equivalence (see [4] for details).…”

Applications of the Stone-Čech compactification to free topological groups

“…Her proof cannot easily be extended to deal with F(X). C. Joiner [4] proved Theorem B. In both cases, particularly the latter, the original proofs are significantly longer and more difficult than ours.…”