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Title: Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF (English)
Author: Herrlich, Horst
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 43
Issue: 2
Year: 2002
Pages: 319-333
Category: math
Summary: The stability of the Lindelöf property under the formation of products and of sums is investigated in ZF (= Zermelo-Fraenkel set theory without AC, the axiom of choice). It is • not surprising that countable summability of the Lindelöf property requires some weak choice principle, • highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, • amusing that finite summability of the Lindelöf property takes place if either some weak choice principle holds or if AC fails drastically. Main results: 1. Lindelöf = compact for $T_1$-spaces iff $\text{\bf CC}(\Bbb R)$, the axiom of countable choice for subsets of the reals, fails. 2. Lindelöf $T_1$-spaces are finitely productive iff $\text{\bf CC}(\Bbb R)$ fails. 3. Lindelöf $T_2$-spaces are productive iff $\text{\bf CC}(\Bbb R)$ fails and $\text{\bf BPI}$, the Boolean prime ideal theorem, holds. 4. Arbitrary products and countable sums of compact $T_1$-spaces are Lindelöf iff $\text{\bf AC}$ holds. 5. Lindelöf spaces are countably summable iff $\text{\bf CC}$, the axiom of countable choice, holds. 6. Lindelöf spaces are finitely summable iff either $\text{\bf CC}$ holds or $\text{\bf CC}(\Bbb R)$ fails. 7. Lindelöf $T_2$-spaces are $T_3$ spaces iff $\text{\bf CC}(\Bbb R)$ fails. 8. Totally disconnected Lindelöf $T_2$-spaces are zerodimensional iff $\text{\bf CC}(\Bbb R)$ fails. (English)
Keyword: axiom of choice
Keyword: axiom of countable choice
Keyword: Lindelöf space
Keyword: compact space
Keyword: product
Keyword: sum
MSC: 03E25
MSC: 54A35
MSC: 54B10
MSC: 54D20
MSC: 54D30
idZBL: Zbl 1072.03029
idMR: MR1922130
Date available: 2009-01-08T19:22:19Z
Last updated: 2012-04-30
Stable URL:
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