Uniformly approachable maps

by

Krzysztof Ciesielski & Dikran Dikranjan

Topology Proc. 20 (1995), 75-89.

Let X and Y be metric spaces. The goal is to study some intermediate classes of functions between the class $C$uc(X,Y) of all uniformly continuous mappings (briefly, UC) from X into Y and the class C(X,Y) of all continuous functions f:X->Y. These classes, defined below, have been intensively studied in earlier papers mainly in the case when Y=R, the real line. In this paper we study them for general Y. In particular, we will consider the case when X=Y is the complex plane C.

1. For subsets K,M of X we say that g:X->Y is a (K,M)-approximation of f if g is a UC map such that g[M] is a subset of f[M] and g(x)=f(x) for each x from K.
2. Function f is uniformly approachable (briefly, UA) if f has a (K,M)-approximation for every compact subset K of X and every subset M of X.
3. Function f is weakly uniformly approachable (briefly, WUA) if f has a (x,M)-approximation for every x form X and every subset M of X.

The class of all uniformly approachable (weekly uniformly approachable, respectively) functions f:X->Y is denoted by $C$ua(X,Y) ($C$wua(X,Y), respectively.) We notice that

(*) $C$uc(X,Y)-->Cua(X,Y)--> Cwua(X,Y)-->C(X,Y),

where the arrow --> stands for the inclusion. Between other things we discuss possible equations in (*) for different spaces X and Y.