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Do we need to provide a composite function when defining a composition of two homeomophisms?

That is,clearly (0,1) is homeormophic to (a,b) and R is homeomorphic to (0,1).In these cases I define a function for each case and show that they are bijective and bi-cts. So what I'm trying to do is defining a composition of these two homeomorphisms,so that R is homeomorphic to (a,b).

So in that case,do I need to provide a function as I do for the above homeomorphisms?

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    yes,composition is also homeomorphism. but is it just enough to say that the composition is homeomorphic of the given two homeomorphism or do I need to provide a function defined on that composition?2012-11-25

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If $f:(a,b)\to (0,1)$ and $g:(0,1)\to\Bbb R$ are homeomorphisms then you can define a homeomorphism $h:(a,b)\to\Bbb R$ by $h=g\circ f$. If you are asking whether you should provide a formula or something for $h$, the answer is no (at least after you know that the composition of homeomorphisms is a homeomorphism, since homeomorphism is an equivalence relation).