Nice
set theoretic formalization of the concept of convergence to points.
Metric spaces are topological spaces. But to put a metric on a
blank topological space, you first have to make it a
uniform space
and then be lucky that it has a
countable basis. Lots of important
topological spaces which cannot be endowed with a metric can be found amoung the
locally convex vector spaces, e.g. the space of generalized functions from the theory of
partial differential equations.