Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF
Herrlich, Horst
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 319-333 / Harvested from Czech Digital Mathematics Library
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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 $\text{\bf CC}(\Bbb R)$, 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.

Publié le : 2002-01-01
Classification:  03E25,  54A35,  54B10,  54D20,  54D30 
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     author = {Horst Herrlich},
     title = {Products of Lindel\"of $T\_2$-spaces are Lindel\"of -- in some models of ZF},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {319-333},
     zbl = {1072.03029},
     mrnumber = {1922130},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119322}
}

Herrlich, Horst. Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 319-333. http://gdmltest.u-ga.fr/item/119322/

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