One place where -subsets occur is when looking at continuous maps from an arbitrary topological space to a metric space (or, more generally, a first countable space). In particular, when considering continuous real-valued functions. Thus we have the following connections to the separation axioms.
In a completely regular space, every singleton set that is a -set is the unique global maximum of a continuous real-valued function.
One direction is obvious. For the other, let be a point in a completely regular space such that is a -set. Let be a sequence of open sets such that . We now define a sequence of functions recursively with the properties:
Having defined , we define as follows. Since is a neighbourhood of and is completely regular, there is a continuous function with support in this neighbourhood and such that . We then compose with a continuous, increasing surjection which maps to . The resulting function is the required .
We then define a function by
By construction, .
We need to prove that this is continuous. First, note that if then for and if then for . Hence the preimage under of is and restricted to this preimage is a scaled translate of . From this, we deduce that the preimage of any open set not containing is open. Thus is continuous everywhere except possibly at . Continuity at is similarly simple: given a set of the form then there is some such that , whence contains all points such that for , which by construction is a neighbourhood of . Hence is continuous and has a single global maximum at .