The axioms of a topological space are usually stated in the "open set" form: A topological space $X$ has a set of subsets $\tau$ whose members satisfy:
- $\emptyset$ and $X$ are in $\tau$.
- $\tau$ is closed under arbitrary unions.
- $\tau$ is closed under finite intersections.
We will call this the usual open set axiomatization of a topological space. The open set axiomatization is probably the most economical, but of course there are other ways as well, such as the Kuratowski closure axioms which axiomatizes the topology using the closure operator.
In his note here, Pete L. Clark gives several alternative characterizations of a topology, most of which have details given. It was mentioned, I think originally by Willard, that it is "possible, but unrewarding to characterize a topology completely by its [boundary]". I could find no further references on this, and I am wondering how it might be done (unrewarding as it may be).
So my question is this: How can a topological space by axiomatized using the boundary operation as the primitive notion? A complete answer should give a list of axioms required for the formulation, as well as a sketch of how they are equivalent to the usual open set formulation.