Products of Lindelöf $T_2$-spaces are Lindelöf – in some models of ZF

Herrlich, Horst

Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 319-333.

Résumé

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.

Détail

Classification : 03E25, 54A35, 54B10, 54D20, 54D30
Keywords: axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum