Let and be open sets in and let be a one-to-one mapping from onto (so that there is an inverse mapping ). Suppose that and are both continuous. Show that for any set whose closure is contained in we have .
I am completely lost. I am assuming that the way to go about this is to show that one is a subset of the other but other than that I have no idea how to proceed.
Edit: So here is my so far very vague attempt or at least idea of how this proof might lead to: If , then so for all and . If this is the correct start to the proof, I am guessing that the continuity of the mappings comes into play but again, I'm lost.