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Excuse the spam, but I would like to ask something which I read in Armstrongs Basic Topology which I am just not 100% sure about.

He says we wish to define homeomorphism such that a circle cannot be homeomorphic to an interval such as [0,1). A continuous function f : X \mapsto Y is one whose every inverse f^-1(N) (N neighbourhood of a mapped point f(x)) is a neighbourhood in X.

He presents a one-to-one and onto function from [0,1) to the circle

x \mapsto exp ( 2∏i x )

This maps the interval into all of the circle. It has an inverse. Exactly why can one not get a continuous inverse??

I suspect this has something to do with the multi-valued property that the complex LOGARITHM suffers from. But somehow I feel the author does not presume the reader to know this much complex analysis - and that there is a more "primitive" reasons dealing with open sets, neighbourhoods, inverses etc. Is there one? Or is the solution merely that there exists no inverse which is not the logarithm?

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# Why the circle can't be homeomorphic to a real interval

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