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Two classes of Fréchet-Urysohn spaces

Author: Alan Dow
Journal: Proc. Amer. Math. Soc. 108 (1990), 241-247
MSC: Primary 54E35; Secondary 03E35, 03E75, 54A35
MathSciNet review: 975638
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Abstract: Arhangel’skii introduced five classes of spaces, ${\alpha _i}$-spaces $\left ( {i < 5} \right )$, which are important in the study of products of Fréchet-Urysohn spaces. For each $i < 5$, each ${\alpha _i}$-space is an ${\alpha _{i + 1}}$-space and it follows from the continuum hypothesis that there are countable ${\alpha _{i + 1}}$-spaces which are not ${\alpha _i}$-spaces. A $v$-space ($w$-space) is a Fréchet-Urysohn ${\alpha _1}$-space (${\alpha _2}$-space). We show that there is a model of set theory in which each ${\alpha _2}$-space ($w$-space) is an ${\alpha _1}$-space ($v$-space).

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Keywords: Fr&#233;chet-Urysohn space, <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$v$">-space, <IMG WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$w$">-space
Article copyright: © Copyright 1990 American Mathematical Society