Is there a natural topology on the set of open sets?

Can one reasonably construct an argument that makes a statement about a set's elements based on matched cardinality without taking into account that sets of equal cardinality aren't necessarily themselves equal?

• Sets of equal cardinality aren't necessarily themselves equal, so that it is possible to construct two sets of equal cardinality with one having an infinite number of elements more than the other. Can one reasonably construct an argument that makes a statement about a set's elements based on matched cardinality (i.e., Cantor's Diagonalization Proof) without taking the aforementioned caveat into account? I get the concept of mapping but, because Cantor's proof deals with "missing elements" (the SoSoS_{o} sequence), it irks me that nothing in the proof addresses the fact that though the sets have equal cardinality and can be mapped one-to-one (because of the vagueness of infinite operations on infinite sets), they are not necessarily equal! Which means that Cantor's proof could just be a very clever tautology, telling us that there are more elements in one set than another set with less elements. Even if you can create an infinite number of SoSoS_{o}-like  sequences, there could be an infinite number of elements missing (but the cardinality is still preserved) that account for them. This question is a follow-up to "".

• Answer:

  If two sets have the same cardinality, then neither one has more elements than the other.  That's the definition of having the same cardinality.