Spaces of urelements (Q1081151)

Without the axiom of choice, topologists have new worlds to explore. Here, the author shows readers familiar with Mostowski's model (described in \textit{U. Felgner}'s ''Models of ZF-set theory'', Lect. Notes Math. 223 (1971; Zbl 0269.02029), that in this model, a Hausdorff space is anti- anti-compact iff it is a continuous one-to-one image of a Dedekind-finite subset of \(U^{\omega}\), where U is the set of ''urelements''. \textit{P. Bankston} [Ill. J. Math. 23, 241-252 (1979; Zbl 0405.54003)] called a space anti-anti-compact if each of its infinite subsets has an infinite compact subset. Under the axiom of choice, Hausdorff spaces with that property must be finite.) The author's results also bear on the interdependence of variants of the axiom of choice.