Open and closed maps

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. topology

Proof

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

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