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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
DOI: https://doi.org/10.1090/S0002-9947-1986-0825729-X
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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Keywords: Left separated, <IMG WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\sigma$">-discrete, point-countable base, reflection, Martin’s Axiom, proper forcing
Article copyright: © Copyright 1986 American Mathematical Society