Nonperfect spaces with point-countable bases

Davies, Peter

doi:10.1090/S0002-9939-1979-0542097-3

Nonperfect spaces with point-countable bases
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by Peter Davies

Proc. Amer. Math. Soc. 77 (1979), 276-278

DOI: https://doi.org/10.1090/S0002-9939-1979-0542097-3

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Abstract:

We construct a completely regular space of cardinality ${\aleph _1}$ with a point-countable base, which is not perfect. This answers a question of Fleissner and Reed. We also construct, under the hypothesis ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$, a hereditarily normal space of cardinality ${\aleph _1}$ with a $\sigma$-disjoint base, which is not perfect.

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