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On my textbook, I was given the following conditions for when a continuous mapping is a homeomorphism:

"Let(X,τ) and (Y,τ′) be topological spaces and f a function from X into Y. Then f is a homeomorphism if and only if (i)f is continuous, (ii)f is one-to-one and onto; that is, the inverse function f^−1:Y→X exists, and (iii)f^−1 is continuous"

I was wondering whether the third criteria is even needed. If a mapping satisfies the first two, shouldn't the inverse mapping automatically be continuous?

Bill
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    Consider the identity map from $\mathbb R$ with discetre topology to $\mathbb R$ with usual topology. [BTW this question has been asked many times on MSE]. – Kavi Rama Murthy Jan 15 '21 at 09:40
  • @KaviRamaMurthy . Even if $(Y,\tau')=(X,\tau),$ a continuous bijection $f:X\to X$ can fail to have a continuous inverse. – DanielWainfleet Jan 15 '21 at 09:52
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    @DanielWainfleet Of course. But that requires a non-trivial construction and I wanted to given an immediately verifiable example. – Kavi Rama Murthy Jan 15 '21 at 09:55
  • ok that makes sense. I guess i was thinking too much about standard functions in calculus – Bill Jan 15 '21 at 10:17

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