Are two spaces obtained by a quotient of the other homeomorphic?

Question

I was thinking about an exercise on quotient topology that asks the following:

If $Y$ and $X$ are two topological spaces such that $Y$ is a quotient of $X$ and $X$ is a quotient of $Y$, need $X$ and $Y$ to be homeomorphic?

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I think the answer is, in general, no, because quotient maps between spaces do not give all the requesites of an homeomorphism between spaces. But I don't see any counterexample to this. Could you give me any hint?

Thanks in advance.

Answer 1

The answer is no. $S^1$ is a quotient of $[0, 1]$ by identifying $0$ and $1$. Likewise, $[0, 1]$ is a quotient of $S^1$ by identifying the upper and lower semi-circles together. Yet, $[0, 1]$ and $S^1$ are not homeomorphic.