This is a survey aimed at a general audience on some recent work related to extending some of the most important results in topology (like Urysohn-Tietze), to the setting of `noncommutative topology relative to a noncommutative function algebra'. By the latter term we mean a collection A of operators which is closed under sums and products. We intend to explain all the needed background. We will spend some time on `noncommutative peak sets', and the problem of finding noncommutative functions (that is, elements of A) that have certain behaviours on certain noncommutative sets. In fact we have recently been able to essentially complete this theory.

This is joint work, mostly with Damon Hay, Matt Neal and Charles Read.