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.
