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?

Since the question about concrete topology is a follow up to a question about discrete topology, I'd bet that the question was intended to have concrete replaced with discrete.

It is true that in the discrete topology, the only continuous functions are constant. This is not true with the concrete topology. If X is more than one point, then f(x)=x for all x in X is a continuous, non-constant function in the concrete topology.

Are you sure you read the question? Just before it asks about the concrete topology, it asks to prove that EVERY function X -> X in the discrete topology is continuous. Why would the next question be to show that only constant functions are continuous in the discrete topology? And you're wrong, in the discrete topology it isn't only the constant mappings that are continuous. All functions are. The original poster proved it in the first post.

Thanks LeonhardEuler, I will attempt to look this up when I get a chance. In the meantime, is there any possibility that the authors meant "from concrete to discrete"?

I just re-checked and the wording is exactly like I have it. It is possible that the author meant that and didn't write it. The copy I have is the first printing of the book.

As has already been noted, the paper by Kiehn (linked to by Matt) cites the very problem that got this thread started. Intriguingly, the actual citation (footnote #2) is to page 199 of the book by Bishop and Goldberg. LeonhardEuler, is this also an error, or is there something on page 199 related to this matter?