Hello, I'm reading this book and I've come to a question that has me stumped: I got the first part: since every set in the discrete topology is open, then the inverse image of every open set must be open. Its the second part that's giving me trouble. It seems to me like many functions could be continuous in the concrete topology even if they are not constant. For example, the identity function: The only open sets in the concrete toplogy are the empty set and X. The inverse image of the empty set is the empty set and the inverse image of X is X. So the inverse images of all the open sets are open. I can't seem to find any flaw in this reasoning. Can someone help me out?