Abstract:
A method of working with $f$-continuous functions on mappings is developed. The method is used to derive a constructive proof of Urysohn's Lemma for mappings. A variant of the Brouwer–Tietze–Urysohn theorem for mappings is proved. Functional characterizations are given for the normality properties of mappings. The notion of perfect normality of a mapping, which seems to be the most optimal, is introduced.
Keywords:fiberwise general topology, $f$-continuous mapping, $\sigma$-normal mapping, perfectly normal mapping, Urysohn's Lemma, Brouwer–Tietze–Urysohn theorem, Vedenisov's conditions of perfect normality.
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