One alternative axiomatization of topology is the touching axioms. It's basically the same as the closure axioms, and is in fact equivalent to the open set definition.
Can we define dimension in a simple way from these axioms.
One way obviously is that you could say "a space is dimension $d$ if given a set of sets such that those sets points don't touch points outside their respective sets, and those sets of sets contain all the points of the space, you can take subsets of those sets such that they still satisfy the same property and each point is in at most $d+1$ of these new sets", but that's a bit wordy.
One idea I had (that didn't work) is "A space is dimension $d$ if a point touches at most $d+1$ sets of a partition of the space." That would say that the U.S.A. is 3-dimensional instead of 2-dimensional.
Any other ideas?