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Proof that (a,b) is not homeomorphic to [a,b]. Assume that they are. Thus we have a continuous, continuously invertible function f from [a,b] to (a,b). Now because f is invertible it is bijective thus f(a)=f(b) is false. Let f(a)<f(b) so that a<f(a)<f(b)<b. Not for f to be invertible we must have a c in (a,b) such that f(c) is in (f(b),b) so that f(b)<f(c). Now by the Intermediate Value Theorem there is a d in (a,c) such that f(d)=f(b). Thus f fails to be invertible if f(a)<f(b).

Note by Samuel Queen
4 years, 7 months ago

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I like this proof. Usually an invariant like compactness is used, but this is more Analysis based.

Philip Doi - 2 years, 11 months ago

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The same result holds if f(a)>f(b).

Samuel Queen - 4 years, 7 months ago

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