I was wondering if there was a way of avoiding having to prove individual invariants of isomorphism and/or homeomorphism are, in fact, invariants. Consider homeomorphisms. We have to prove that compactness is a topological property. The wikipedia page on topological properties states that "Informally, a topological property is a property of the space that can be expressed using open sets." Is there a way to formalize this? Since a homeomorphism preserve the structure of open sets, it seems any property "formulated in terms of open sets" must immediately be invariant.
The situation is similar with isomorphisms. Consider a vector space isomorphism. We need to prove (fairly easily) that isomorphic vector spaces have the same number (cardinality) of dimensions. It seems we should be able to say something like "any property of vector space phrased in terms of the vector space structure is preserved under isomorphism".