Products of Lindelöf \(T_2\)-spaces are Lindelöf – in some models of ZF. (English) Zbl 1072.03029

Herrlich, Horst

Commentat. Math. Univ. Carol. 43, No. 2, 319-333 (2002).

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
(1) not surprising that countable summability of the Lindelöf property requires some weak choice principle, (2) highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, (3) amusing that finite summability of the Lindelöf property takes place if either some weak choice principle holds or if AC fails drastically.
Some results:
1. Lindelöf = compact for \(T_1\)-spaces iff \(\text{CC}(\mathbb R)\), the axiom of countable choice for subsets of the reals, fails.
2. Lindelöf \(T_1\)-spaces are finitely productive iff \(\text{CC}(\mathbb R)\) fails.
3. Lindelöf \(T_2\)-spaces are productive iff \(\text{CC}(\mathbb R)\) fails and BPI, the Boolean prime ideal theorem, holds.

MSC:

03E25	Axiom of choice and related propositions
54A35	Consistency and independence results in general topology
54B10	Product spaces in general topology
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

axiom of choice; axiom of countable choice; Lindelöf space; compact space; product; sum