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