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K. Tamano asked the following question. Is a stratifiable space metrizable if it is K-metrizable? To answer this question, we show that the class of stratifiable K-metrizable spaces is much wider than that of metrizable spaces. In fact, we describe two very distinct classes of countable, stratifiable K-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 K-metrizability of countable spaces defined by the Cantor tree and note a topological property of monotonically normal Kmetrizable spaces.

Countable size nonmetrizable spaces which are stratifiable and 𝜅-metrizable