Free topological groups on metrizable spaces and inductive limits

✍ Scribed by Vladimir Pestov; Kohzo Yamada

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 107 KB
• Volume: 98
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00103-5

✦ Synopsis

We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {A n (X): n ∈ N}, where A n (X) is formed by all words of reduced length n; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F (X) is the inductive limit of the sequence {F n (X): n ∈ N}; (ii) X is either locally compact separable or discrete.