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Free topological groups and inductive limits

✍ Scribed by Michael G. Tkačenko


Book ID
107913870
Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
882 KB
Volume
60
Category
Article
ISSN
0166-8641

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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

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A topological group G is sequentially complete if it is sequentially closed in any other topological group. We show that for a Tychonoff space X, the free topological group F (X) is sequentially complete iff the free Abelian topological group A(X) is sequentially complete iff X is sequentially close