1. The problem statement, all variables and given/known data The problem is as follows: 2. Relevant equations We are using the definition that D is dense if its closure is the whole space. Proofs using this definitions would be best as we were not taught any equivalent ones. Not sure if relevant but just in case: ~ is an equivalence relation; ~ topologies share dense sets; ~ topologies share sets with the empty interior 3. The attempt at a solution I started (a) by saying T1 and T2 share dense sets and assuming they aren't ~ but I have yet to see anyway to reach a contradiction. I also thought about trying a contra-position but I'm not sure it would pay off either. I'm pretty clueless about how to approach (b) altogether. I would appreciate any help you can offer even if it just get me started. These are actual the last questions in a set of questions and I already managed to get the earlier ones these I just can't see. Thanks for any help.