[xanderlewis]

No, they are not.

In the usual mathematical sense of the words you are using, topologies aren’t even the right type of object to admit a notion of continuity. Your statement doesn’t even make sense. It’s maps between them that can be continuous.

In fact, a topological space is sort of the minimal amount of structure a set needs to have to be able to talk about continuity of maps to/from it.

[mathgradthrow]

It is not always done, but it is still correct, to replace the objects of a category with the identity morphisms. So in Top, it is totally correct to think of topological space as the identity homeomorphism, which is indeed continuous.

[xanderlewis]

I'm aware; in mathematics it's possible to replace almost anything with some other thing to make the statement you want to be true come true. But it's usually just gonna confuse everyone.

[mathgradthrow]

Well the thing I chose to replace the thing with is actually isomorphic (type equivalent) to the thing I replaced. So that's quite a bit more constrained than "replacing anything with anything". Not only are the arrows the only thing that matters, but its cleaner to suppose that they're the only thing there is.