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We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of F, subsets of normal spaces.Mathematics Subject Classification: 03325, 04A25, 54D10, 54D15.

Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.Mathematics Subject Classification: 03E25, 04A25, 54B10.

We study the relationship between various properties of the real numbers and weak choice principles.Mathematics Subject Classification: 03E25, 04A25.

Abstract. The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.Mathematics Subject Classification: 03E25, 04A25, 54D10, 54D15.