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On \(\Theta\)-regular spaces. (English) Zbl 0809.54017
Summary: We study \(\theta\)-regularity and its relations to other topological properties. We show that the concepts of \(\theta\)-regularity [D. S. Janković, ibid. 8, 615-619 (1985; Zbl 0577.54012)] and point paracompactness [J. M. Boyte, J. Aust. Math. Soc. 15, 138-144 (1973; Zbl 0269.54011)] coincide. Regular, strongly locally compact or paracompact spaces are \(\theta\)-regular. We discuss the problem when a (countably) \(\theta\)-regular space is regular, strongly locally compact, compact, or paracompact. We also study some basic properties of subspaces of a \(\theta\)-regular space. Some applications: A space is paracompact iff the space is countably \(\theta\)-regular and semiparacompact. A generalized \(F_ \sigma\)-subspace of a paracompact space is paracompact iff the subspace is countably \(\theta\)-regular.

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54B05 Subspaces in general topology
54D45 Local compactness, \(\sigma\)-compactness
54D30 Compactness
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