A bijections is open iff it is closed

Let $𝑓 : 𝑋 ⤖ 𝑌$ be a bijection between topological spaces. Then $𝑓$ is open iff it is closed. #m/thm/topology

Proof

Let $𝑓$ is closed iff for every open $𝑈 \subseteq 𝑋$ , $𝑓 (𝑋 ∖ 𝑈) = 𝑌 ∖ 𝑉$ for some open $𝑉 \subseteq 𝑌$ , which by bijectivity is true iff $𝑓 (𝑈) = 𝑉$ .

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