Geodesic
First Countable Spaces that Have Special Pseudo-Bases
Canadian mathematical bulletin, Tome 21 (1978) no. 1, pp. 103-104

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Two types of pseudo-bases, σ-disjoint and σ-discrète, are utilized in this note. In the next section, we show that a first countable Hausdorff space has a σ-disjoint pseudo-base if and only if it has a dense metrizable subspace. This result implies that many first countable spaces have dense metrizable subspaces. In section 3, we show that if X is a Hausdorff space that either is quasi-developable or has a base of countable order, then X has a dense metrizable subspace if and only if it has a dense metrizable Gδ subspace. We give an example to show that the conclusion of this theorem is false for semi-metrizable spaces. Finally, in the last section, we investigate when a quasi-developable (resp. semi-metrizable) space can be embedded as a dense subspace of a quasi-developable (resp. semi-metrizable) Baire space. 
Jr, H. E. White. First Countable Spaces that Have Special Pseudo-Bases. Canadian mathematical bulletin, Tome 21 (1978) no. 1, pp. 103-104. doi: 10.4153/CMB-1978-016-5
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