Weakly complete free topological groups

✍ Scribed by Dikran Dikranjan; Michael Tkačenko

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 269 KB
• Volume: 112
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00235-7

✦ Synopsis

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 closed in βX. Furthermore, the free precompact Abelian group F (X, PA) is sequentially complete iff the space X is sequentially closed in βX. We consider also other forms of weak completeness, namely ω-completeness and k-completeness, introduced analogously by means of the ω-closure and the k-closure. We prove that the groups A(X) and F (X) are ω-complete (k-complete) iff X is ωclosed (k-closed) in the Dieudonné completion µX of X.