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\(\alpha_4\)-property versus \(A\)-property in topological spaces and groups. (English) Zbl 0902.22001
The notions of \(\alpha_4\)-space and \(A\)-space were introduced by A. Arhangel’skii (1972) and E. Michael (1973), respectively. The main results of the paper are the following:
1. If \(X\) is a regular sequential space such that each point of \(X\) is a \(G_\delta\)-set, then \(X\) is an \(\alpha_4\)-space if and only if \(X\) is an \(A\)-space.
2. If \(G\) is a sequential topological group such that either \(e\in G\) is a \(G_\delta\)-set or \(G\) is hereditarily normal, then the following conditions are equivalent:
a) \(G\) is an \(\alpha_4\)-space,
b) \(G\) is an \(A\)-space,
c) \(G\) is strongly Fréchet.

MSC:
22A05 Structure of general topological groups
54A05 Topological spaces and generalizations (closure spaces, etc.)
54H11 Topological groups (topological aspects)
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