What are and ? The usual and cofinite topologies? And I assume .

I agree that it's closed. If then and so is open. The image of any subset of is closed.

I also agree that it can't be continuous. For, if were continuous then where is the subspace topology would be continuous. But, every subspace of a cofinite space has the cofinite topology, and a finite cofinite space is discrete. And so, where is the two point discrete space.

Thus, if were continuous there would be a map which is surjective and continuous. What's the problem with that?