Vol. 55, No. 2, 1974

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Metrizability of topological spaces

Richard Earl Hodel

Vol. 55 (1974), No. 2, 441–459

This paper is a study of conditions under which a topological space is metrizable or has a countable base. In § 2 we consider the metrizability of spaces having a weak base in the sense of Arhangel’skiǐ. In § 3 we extend earlier work of Bennett on quasi-developments by showing that every regular 𝜃-refinable β-space with a quasi-Gδ-diagonal is semi-stratifiable. One consequence of this result is a generalization of the Borges-Okuyama theorem on the metrizability of a paracompact wΔ-space with a Gδ-diagonal. In § 4 we prove that a regular space has a countable base if it is hereditarily a CCC wΔ-space with a point-countable separating open cover. This result is motivated by the remarkable theorem of Arhangel’skiǐ which states that a regular space has a countable base if it is hereditarily a Lindelöf p-space. In § 5 we show that every regular p-space which a Baire space has a dense subset which is a paracompact p-space. This result, related to work of Šapirovskiǐ, is then used to obtain conditions under which a Baire space satisfying the CCC is separable or has a countable base. In § 6 we prove that every locally connected, locally peripherally separable meta-Lindelöf Moore space is metrizable. Finally, in § 7 we consider the metrizability of spaces which are the union of countably many metrizable subsets. The results obtained in this section extend earlier work of Čoban, Corson-Michael, Smirnov, and Stone.

Mathematical Subject Classification 2000
Primary: 54E35
Received: 7 March 1974
Published: 1 December 1974
Richard Earl Hodel