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  • $\begingroup$ By my last sentence I mean, to undergraduates who have not yet seen what a topological space is! $\endgroup$ Commented Mar 23, 2010 at 23:56
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    $\begingroup$ David: That's exactly right. What you write is interesting, but I think it's clear that an answer of the form `The axioms for a topological space are equivalent to these other ones.' is not quite satifactory. $\endgroup$ Commented Mar 24, 2010 at 0:12
  • $\begingroup$ I like this answer. $\endgroup$ Commented Mar 25, 2010 at 8:40
  • $\begingroup$ "the category of topological spaces is equivalent to the category of relational algebras for the ultrafilter monad." -- they are isomorphic, I guess. $\endgroup$ Commented Mar 30, 2015 at 9:21