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.

For a compactification αX of a Tychonoff space X, the algebra of all functions f ∈ C(X) that are continuously extendable over αX is denoted by C α (X). It is shown that, in a model of ZF, it may happen that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that C α (X) = C γ (X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX × βY is the Čech-Stone compactification of X × Y when X × Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C *embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a functional Čech-Stone compactification is investigated in the absence of the axiom of choice.

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