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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)