- Let X & Y be topological spaces and let f: X --> Y. Suppose A U B = X. And that f limited to A gives: A ---> Y and f limited to B gives: B ---> Y and these are continuous.
Basically with this problem, I need to show that f is continuous if A and B are open and if A and B are closed. My initial thoughts are that in the first case X must be open since unions of open sets are open. My question is that am I allowed to assume open sets exist in Y? Because then I can feasibly apply the topological continuity theorem (it's called theorem 2.6 in Basic Topology by M.A. Armstrong) which essentially says that the preimage of open sets are open. And I think that would prove it is continuous. The second case troubles me because I don't think I can say conclusively that the preimage of closed sets are closed. I know that if X is closed then X complement must be open. But I'm pretty sure X complement is not in the domain of f. Apparently it's also possible to find a function such that given these constraints, the function is not continuous. Any help on this is greatly appreciated. Thanks!