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Expandabilities of product spaces. (English summary) Topology Proc. 37 (2011), 95–106. In what follows, all spaces will be T1-spaces, and if {Gξ | ξ ∈ Ξ} is a family of subsets of a set X, we will assume that Gξ = Gη whenever ξ = η. Recall that a space X is called expandable (resp., discretely expandable) if for every locally finite (resp., discrete) family {Fξ | ξ ∈ Ξ} of subsets of X there exists a locally finite family {Gξ | ξ ∈ Ξ} of open subsets of X such that Fξ ⊆ Gξ for each ξ ∈ Ξ. The notions of σ-expandability and discrete σ-expandability were introduced by P. Y. Zhu [Sci. Math. Jpn. 65 (2007), no. 2, 173– 178; MR2299202 (2007k:54016)], and the notions of (discrete) θ-expandability and (discrete) subexpandability by Y. Katuta [Fund. Math. 87 (1975), no. 3, 231–250; MR0377817 (51 #13986)]. In this paper, the author proves that (1) if X is a normal P-space (in the sense of K. Morita [Math. Ann. 154 (1964), 365–382; MR0165491 (29 #2773)]), Y is a paracompact Σ-space (in the sense of K. Nagami [Fund. Math. 65 (1969), 169–192; MR0257963 (41 #2612)]) and X is σ-expandable (resp., θ-expandable), then so is X × Y; (2) if X is a normal P-space, Y is metrizable and X is discretely σ-expandable (resp., discretely θ-expandable; discretely subexpandable; subexpandable)

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86e, 54030

Year: 2013

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