A method of working with f -continuous functions on mappings is developed. The method is used to derive a constructive proof of Urysohn 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.
M. Yu. Liseev (Fri,) studied this question.