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We study spaces with sharp bases and bases of countable order. A characterization of spaces with external bases of countable order is established (Theorem 2.7). Some necessary and sufficient conditions for a space X × S , where S is the convergent sequence, to have a sharp base are given (Theorem 3.2). It follows that a pseudocompact space X is metrizable iff X × S has a sharp base (Corollary 3.3). It is proved that a sharp base of finite rank is a uniform base (Theorem 4.4). Some other new results are also obtained, and some open questions are formulated.

Given a space M, a family of sets A of a space X is ordered by M if A = { A K : K is a compact subset of M} and K ⊂ L implies A K ⊂ A L . We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space C p ( X ) belongs to M if and only if it is a Lindelof Σ-space. Under MA ( ω 1 ) , if X is compact and ( X × X ) \ Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ = { ( x , x ) : x ∈ X } is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X 2 \ Δ belongs to M then X is metrizable in ZFC. We also consider the class M ⁎ of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P ⊂ X there exists F ∈ F with P ⊂ F . It is a ZFC result that if X is a compact space and ( X × X ) \ Δ belongs to M ⁎ then X is metrizable. We also establish that, under CH, if X is compact and C p ( X ) belongs to M ⁎ then X is countable.

We give a new metrization theorem on terms of a new structure introduced by the authors in [Rend. Instit. Mat. Univ. Trieste 30 (1999) 21–30] and called fractal structure. This allows us to approach some classical and new metrization theorems (due to Nagata, Smirnov, Moore, Arhangel'skii, Frink, Borges, Hung, Morita, Fletcher, Lindgren, Williams, Collins, Roscoe, Reed, Rudin, Hanai, Stone, Burke, Engelking and Lutzer) from a new point of view.

Inspired by results in the theory of compact topological groups, a generalization of Katětov's famous metrization theorem: `compact spaces with hereditarily normal cube are metrizable' is presented.

The notions of a weak k -development and of a weak development, defined in terms of sequences of open covers, were recently introduced by the first and the third authors. The first notion was applied to extend in an interesting way Michael's Theorem on double set-valued selections. The second notion is situated between that of a development and of a base of countable order. To see that a space with a weak development has a base of countable order, we use the classical works of H.H. Wicke and J.M. Worrell. We also introduce and study the new notion of a sharp base, which is strictly weaker than that of a uniform base and strictly stronger than that of a base of countable order and of a weakly uniform base, and which is strongly connected to the notion of a weak development. Several examples are exhibited to prove that the new notions do not coincide with the old ones. In short, our results show that the notions of a weak development and of a sharp base fit very well into already existing system of generalized metrizability properties defined in terms of sequences of open covers or bases. Several open questions are formulated.

We show that the General Metrization Problem posed by the advent of Urysohn's Theorem has solutions other than those in Bing's Theorem and its generalizations, and give a theorem that uses no counterpart to Bing's discreteness or (any of its generalizations such as) local finiteness or the closure preserving property or the cushion property. There, metrizability is equated to some sort of Regularity, with the separating open sets (of a closed set and a point) coming in in a specific manner from a specific family.

This is a survey article on the work of Professor Jun-iti Nagata. We present his educational and professional career, his activities and contributions to the topology community and his work on topology, especially the theory of uniform spaces, rings of continuous functions, metrization and generalized metric spaces.

This paper deals with the study of closed images or quasi-perfect images of Nagata spaces, contraconvergent spaces, weak contraconvergent spaces, ks-spaces, γ-spaces and wγ-spaces and, of metrization theorems involving these spaces. We prove that the closed images of contraconvergent (weak contraconvergent) spaces are contraconvergent (weak contraconvergent) and that quasi-perfect images of γ- (wγ-)spaces are γ (wγ).

In this paper we study the role of Stepanova's continuous separating families in the class of linearly ordered and generalized ordered spaces and we construct examples of paracompact spaces that have strong base properties (such as point-countable bases or σ -disjoint bases), have continuous separating families, and yet are non-metrizable.

In this paper we examine the role of the β-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a β-space with a G δ -diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.