A bijective continuous map is a homeomorphism if and only if it is open or closed
by Hidenori
Proposition
Suppose $f: X \rightarrow Y$ is a bijective continuous map. Show that the following are equivalent:
- $f$ is a homeomorphism.
- $f$ is open.
- $f$ is closed.
Solution
$1 \rightarrow 2$
$f$ is a homeomorphism, so it must be open.
$2 \rightarrow 3$
Let $C \subset X$ be closed. Then $f(X \setminus C)$ is open in $Y$ because $f$ is open. Since $f$ is bijective, $f(X \setminus C) = f(X) \setminus f(C)$. Thus $f(C)$ is closed in $Y$.
$3 \rightarrow 1$
Let $U \subset X$ be open. Then $f(X \setminus U)$ is closed in $Y$ because $f$ is closed. Since $f$ is bijective, $f(X \setminus U) = f(X) \setminus f(U)$. Thus $f(U)$ is open in $X$.
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