"A topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets." (I copied it from Wikipedia)
Now my question is: What is the definition of a topological property ? Of course you can define it as wiki defines it. But I am more concerned about the the part of wiki's "definition" which says that "Informally, a topological property is a property of the space that can be expressed using open sets."
Is there a definition of a topological property that says which well formed formulas are well formed formulas of topological properties and which are not ?
Because of what I read in wikipedia, I was expecting to see a definition of a topological property that talks about the internal structure of the well formed formula of the property. Then, I also expected that there was a theorem that says that if $(X_1,T_1),(X_2,T_2)$ are any two homeomorphic topological spaces and the well formed formula $\phi(X,T)$ is a topological property, then:
$\phi(X_1,T_1)$ iff $\phi(X_2,T_2)$
Is there such a definition and such a theorem ?
Such a definition and such a theorem will enable one to spot many topological properties easily.
Here is a similar question: Can you characterize all properties of topological spaces which are preserved by homeomorphisms