This question is motivated by the one here.
I am not familiar with the formalism the OP uses but, in short, it is about whether it is possible to formulate a first-order theory for a single topology or not. One of the paragraphs of the answer provided by Noah Schweber says the following:
This is the sense in which topology is not first-order: any attempt to "first-orderize" topology will result in things which look like topological spaces "up to first-order facts" but are not in fact topological spaces.
I am asking for examples of those things which look like topological spaces up to first-order facts. I would like to know if such things, not only topological spaces but objects which look like common objects we use in maths up to first-order (or higher) facts have ever been used or studied.
PS: There is a tag called ''generalized topology''. I have looked for the definition on Google and it seems it is not related to this, but who knows...
