Horst Herrlich
Products of Lindel\"of $T_2$-spaces are Lindel\"of --- in some models of ZF

Comment.Math.Univ.Carolinae 43,2 (2002) 319-333.

Abstract:The stability of the Lindel\"of 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 \roster \item "$\bullet $" not surprising that countable summability of the Lindel\"of property requires some weak choice principle, \item "$\bullet $" highly surprising, however, that productivity of the Lindel\"of property is guaranteed by a drastic failure of {AC}, \item "$\bullet $" amusing that finite summability of the Lindel\"of property takes place if either some weak choice principle holds or if {AC} fails drastically. \endroster \par \noindent Main results: \roster \item "1." Lindel\"of = compact \hskip 1em\relax for $T_1$-spaces \newline iff $\text {CC}(\Bbb R)$, the axiom of countable choice for subsets of the reals, fails. \item "2." Lindel\"of $T_1$-spaces are finitely productive \newline iff $\text {CC}(\Bbb R)$ fails. \item "3." Lindel\"of $T_2$-spaces are productive \newline iff $\text {CC}(\Bbb R)$ fails and {BPI}, the Boolean prime ideal theorem, holds. \item "4." Arbitrary products and countable sums of compact $T_1$-spaces are Lindel\"of \newline iff {AC} holds. \item "5." Lindel\"of spaces are countably summable \newline iff {CC}, the axiom of countable choice, holds. \item "6." Lindel\"of spaces are finitely summable \newline iff either {CC} holds or $\text {CC}(\Bbb R)$ fails. \item "7." Lindel\"of $T_2$-spaces are $T_3$ spaces \newline iff $\text {CC}(\Bbb R)$ fails. \item "8." Totally disconnected Lindel\"of $T_2$-spaces are zerodimensional \newline iff $\text {CC}(\Bbb R)$ fails. \endroster

Keywords: axiom of choice, axiom of countable choice, Lindel\"of space, compact space, product, sum
AMS Subject Classification: 03E25, 54A35, 54B10, 54D20, 54D30