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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Left separated spaces with point-countable bases


Author: William G. Fleissner
Journal: Trans. Amer. Math. Soc. 294 (1986), 665-677
MSC: Primary 03E35; Secondary 03E55, 54D18, 54E18
MathSciNet review: 825729
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0825729-X
PII: S 0002-9947(1986)0825729-X
Keywords: Left separated, $ \sigma $-discrete, point-countable base, reflection, Martin's Axiom, proper forcing
Article copyright: © Copyright 1986 American Mathematical Society