Products of Lindelöf T 2 -spaces are Lindelöf – in some models of ZF

Horst Herrlich

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 319-333
  • ISSN: 0010-2628

Abstract

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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 CC ( ℝ ) , the axiom of countable choice for subsets of the reals, fails. 2. Lindelöf T 1 -spaces are finitely productive iff CC ( ℝ ) fails. 3. Lindelöf T 2 -spaces are productive iff CC ( ℝ ) fails and BPI , the Boolean prime ideal theorem, holds. 4. Arbitrary products and countable sums of compact T 1 -spaces are Lindelöf iff AC holds. 5. Lindelöf spaces are countably summable iff CC , the axiom of countable choice, holds. 6. Lindelöf spaces are finitely summable iff either CC holds or CC ( ℝ ) fails. 7. Lindelöf T 2 -spaces are T 3 spaces iff CC ( ℝ ) fails. 8. Totally disconnected Lindelöf T 2 -spaces are zerodimensional iff CC ( ℝ ) fails.

How to cite

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Herrlich, Horst. "Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 319-333. <http://eudml.org/doc/249001>.

@article{Herrlich2002,
abstract = {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 $\text\{\bf CC\}(\mathbb \{R\})$, the axiom of countable choice for subsets of the reals, fails. 2. Lindelöf $T_1$-spaces are finitely productive iff $\text\{\bf CC\}(\mathbb \{R\})$ fails. 3. Lindelöf $T_2$-spaces are productive iff $\text\{\bf CC\}(\mathbb \{R\})$ 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 $\text\{\bf CC\}(\mathbb \{R\})$ fails. 7. Lindelöf $T_2$-spaces are $T_3$ spaces iff $\text\{\bf CC\}(\mathbb \{R\})$ fails. 8. Totally disconnected Lindelöf $T_2$-spaces are zerodimensional iff $\text\{\bf CC\}(\mathbb \{R\})$ fails.},
author = {Herrlich, Horst},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum; axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum},
language = {eng},
number = {2},
pages = {319-333},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF},
url = {http://eudml.org/doc/249001},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Herrlich, Horst
TI - Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 319
EP - 333
AB - 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 $\text{\bf CC}(\mathbb {R})$, the axiom of countable choice for subsets of the reals, fails. 2. Lindelöf $T_1$-spaces are finitely productive iff $\text{\bf CC}(\mathbb {R})$ fails. 3. Lindelöf $T_2$-spaces are productive iff $\text{\bf CC}(\mathbb {R})$ 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 $\text{\bf CC}(\mathbb {R})$ fails. 7. Lindelöf $T_2$-spaces are $T_3$ spaces iff $\text{\bf CC}(\mathbb {R})$ fails. 8. Totally disconnected Lindelöf $T_2$-spaces are zerodimensional iff $\text{\bf CC}(\mathbb {R})$ fails.
LA - eng
KW - axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum; axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum
UR - http://eudml.org/doc/249001
ER -

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