On \(\Theta\)-regular spaces (Q1338421)

Summary: We study \(\theta\)-regularity and its relations to other topological properties. We show that the concepts of \(\theta\)-regularity [\textit{D. S. Janković}, ibid. 8, 615-619 (1985; Zbl 0577.54012)] and point paracompactness [\textit{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.