Why is the class of topological spaces not axiomatizable? This is a follow-up question to this one, where in the answer it is explained how topological spaces may very well be described in a purely first-order manner. Furthermore, the set of first-order sentences that hold in every topological space seems to include equivalents of the axioms of topology. On the other hand it is stated - as a key fact - that the class of topological spaces is not (first-order) axiomatizable. At first sight, this seems paradoxical. 

Is there a simple proof of the
  nonaxiomatizability of topology?

References are welcome!

Although the discreteness of the space is expressible in the set-up proposed by Carl's answer to your other question (where the open sets are treated as first-order objects with their own sort), nevertheless the class of infinite discrete topological spaces will not form an elementary class. The reason is that it has no countable models. Once there are infinitely many points, then by discreteness we know that the open sets will consist of all subsets of those points, and this forces the model to have at least continuum many open sets. This would violate the downward Löwenheim-Skolem theorem if it were an elementary class.  
