Countable size nonmetrizable spaces which are stratifiable and 𝜅-metrizable

Abstract

K. Tamano asked the following question. Is a stratifiable space metrizable if it is κ \kappa -metrizable? To answer this question, we show that the class of stratifiable κ \kappa -metrizable spaces is much wider than that of metrizable spaces. In fact, we describe two very distinct classes of countable, stratifiable κ \kappa -metrizable spaces which are not metrizable. One of them has no nontrivial convergent sequence. The other is bisequential but not a w-space. In addition, we give a characterization of κ \kappa -metrizability of countable spaces defined by the Cantor tree and note a topological property of monotonically normal κ \kappa -metrizable spaces.