# If there are continuous surjections $f: X \to Y$ and $g: Y \to X$, are $X, Y$ homeomorphic? [duplicate]

If there are continuous surjections $f: X \to Y$ and $g: Y \to X$ are $X, Y$ homeomorphic? This is not homework. Just to trying to review some topology.

Hint: Let $X=\mathbb{S}^1$ and $Y=[0,1]$.