The problem of extending partial functions

Abstract

There are several general concepts and ideas in contemporary mathematics which play a fundamental role in almost all of its branches. Among ideas of this kind, the concept of extending a given partial function is of undoubted interest and of paramount importance for various domains of mathematics. For instance, every working mathematician knows that the problemof extending partial functions is considered and intensively studied in universal algebra, general and algebraic topology, mathematical and functional analysis, as well as other fields. Of course, this problem has specific features in any of the above-mentioned disciplines and it frequently needs special approaches or appropriate research tools which are suitable only for a given situation and are applicable to concrete mathematical objects, for example, groups, topological spaces, ordered sets, differentiable manifolds, and other structures.