Surely sets with the same cardinality are homeomorphic if we assign both of them the discrete topology. What's preventing us from doing that? For example, (0,1) and (2,3) \cup (4,5) have the same cardinality. With the natural subspace topology they are not homeomorphic - as one is connected and the other isn't. However I could say that they are homeomorphic by assigning the discrete topology on both. Why can't we do that? Is it because that we would lose our intuition of 'continuous deformation' if we did this? This may sound stupid but I can't seem to get my head around this. Thanks.