Left separated spaces with point-countable bases

Fleissner, William G.

doi:10.1090/S0002-9947-1986-0825729-X

Left separated spaces with point-countable bases
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Trans. Amer. Math. Soc. 294 (1986), 665-677 Request permission

Abstract:

Theorem 2.2 lists properties equivalent to left separated spaces in the class of ${T_1}$ with point-countable bases, with examples preventing plausible additions to this list. For example, $X$ is left iff $X$ is $\sigma$-weakly separated or $X$ has a closure preserving cover by countable closed sets, but $X$ is left separated does not imply that $X$ is $\sigma$-discrete. Theorem 2.2 is used to show that the following reflection property holds after properly collapsing a supercompact cardinal to ${\omega _2}$: If $X$ is a not $\sigma$-discrete metric space, then $X$ has a not $\sigma$-discrete subspace of cardinality less than ${\omega _2}$. Similar reflection properties are shown true in some models and false in others.

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Combinatorial set theory