locally

Let $\mathfrak M$ be a class of mappings between compact spaces. A surjective mapping $f: X \to Y$ between continua is said to be locally $\mathfrak M$ provided that for each point $x \in X$ there is a closed neighborhood V of x such that f(V) is is a closed neighborhood of f(x) and that the restriction f|V is in $\mathfrak M$ (see [59, Chapter 4, Section C, p. 18]).