Countable sums and products of Loeb and selective metric spaces

Horst Herrlich; Kyriakos Keremedis; Eleftherios Tachtsis

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Disasters in metric topology without choice

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We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X, d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply .

Tychonoff products of super second countable and super separable metric spaces

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Complete Spaces

Karol Pąk (2008)

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This paper is a continuation of [12]. First some definitions needed to formulate Cantor's theorem on complete spaces and show several facts about them are introduced. Next section contains the proof of Cantor's theorem and some properties of complete spaces resulting from this theorem. Moreover, countable compact spaces and proofs of auxiliary facts about them is defined. I also show the important condition that every metric space is compact if and only if it is countably compact. Then...