Abstract. We continue our investigations into absolute CR -epic
spaces. Given a continuous function f : X → Y , with X absolute
CR -epic, we search for conditions which imply that Y is also absolute CR -epic. We are particularly interested in the cases when X is
a dense subset of Y and when f is a quotient mapping. To answer
these questions, we consider issues of local extension of continuous
functions. The results on this question are of independent interest.