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Keywords:
$\aleph_1$-calibre; star countable; zeroset diagonal
Summary:
We prove that, assuming \emph{CH}, if $X$ is a space with $\aleph_1$-calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_\delta$-diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469--473. We also make some observations on spaces with $\aleph_1$-calibre.
References:
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