  The goal of this paper is to generalise Alex Rennet's proof of the
non-axiomatizability of the class of pseudo-o-minimal structures. Rennet showed
that if L is an expansion of the language of ordered fields and K is the class
of pseudo-o-minimal L-structures (L-structures elementarily equivalent to an
ultraproduct of o-minimal structures) then K is not computably axiomatizable.
We give a general version of this theorem, and apply it to several classes of
topological structures.

Logic