I'm studying general topology and a question has come to my mind.
We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.
I can see the ambiguity of this definition lying in its lacking to specify the language in which the properties are expressed. Anyway, I was wondering if, in some appropriate language, topological properties are enough to capture the notion of homeomorphisms.
More precisely, is it true that, if two spaces satisfy the same topological properties written in an appropriate (formal) language, then they are homeomorphic? Feel free to make assumptions on the language of the properties.
Disclaimer If we think about properties in the most general and informal sense, then the answer is yes. Indeed, given a topological space X, "being homeomorphic to X" is a topological property. As a result, given another space Y having the same topological properties as X, it is indeed homeomorphic to X. The question may get more interesting restricting the language in which properties can be expressed.
Best Answer
since first order theories with infinite models always have elementarily equivalent but non-isomorphic models, one has to look for something else. hen:
if you're willing to allow for one infinitary operation, the theory of complete heyting algebras is a good first approximation: every topological space provides a model, and there are ways to express many properties one can regard as 'topological', though not all models 'are' (or rather, 'are provided by') spaces, and there are some issues involving separation properties, so that some (non-hausdorff) non-homeomorphic spaces end up being 'elementarily equivalent'
another possible approach is via a relational theory of ultrafilter convergence, which presumably can be developed in two-sorted FOL