Topological properties are investigated because we can show that two spaces are not homeomorphic by finding one property that holds in one space but not the other. But what if no topological property can distinguish two topological spaces? So I ask:
If two topological spaces have the same topological properties, must they be homeomorphic?
Edit: I don't really have any particular class of topological properties in mind, because what I am thinking is really every single topological property, whenever it is well-defined for a topological space. I just didn't know that the class of topological properties is so large, that even "homeomorphic to a space $X$" is itself a topological property, making my question trivial.