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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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