Functional approach to the normality of mappings

In the article a technique of the usage of $f$-continuous functions (on mappings) and their families is developed. A proof of the Urysohn’s Lemma for mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension Theorem for mappings is proven. Characterizations of the normality properties of mappings are given and the notion of a perfect normality of a mapping is introduced. It seems to be the most optimal in this approach.