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 ArhangeΓskiϊ. In § 3 we extend earlier work of Bennett on quasi-developments by showing that every regular 0-refinable β-space with a quasi-G 5 -diagonal is semi-stratifiable. One consequence of this result is a generalization of the Borges-Okuyama theorem on the metrizability of a paracompact wi-space with a

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... G^diagonal. In § 4 we prove that a regular space has a countable base if it is hereditarily a CCC wA-space with a point-countable separating open cover. This result is motivated by the remarkable theorem of ArhangeΓskiϊ which states that a regular space has a countable base if it is hereditarily a Lindelδf 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 Sapirovskiϊ, 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-Lindelδf 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 Coban, Corson-Michael, Smirnov, and Stone. !• Preliminaries* We begin with some definitions and known results which are used throughout this paper. Unless otherwise stated, no separation axioms are assumed; however, regular, normal, and collectionwise normal spaces are always TΊ and paracompact spaces are always Hausdorff. The set of natural numbers is denoted by N, and i f j, k, m, n, r, s, and t denote elements of N. Let X be a set, let ^ be a collection of subsets of X, let p be an element of X.