Need help in some inter-dimensional isomorphisms

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Need help in some "inter-dimensional isomorphisms"

consider the set
M = {e^(i*arctan(x)) in C | x in R }

now it is obvious that M is isomorphic to the real line, so we have an isomorphism from a subset of 2D to 1D.
ok, now we should have M x M isomorphic to R^2, but somehow I cannot do this rigorously (excuse the spelling? :)
what I do know (if there is no mistake in my working :) is that M x M is a hollow torus (doughnut :) in R^4 or C^2 if one allows x to be infinity in the definition

if there are anyone willing to help - i will nbe greatly indebted


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If you change the definition of M (by "allowing x to go to infinity"), then why would you think M is still isomorphic to R?

(And P.S. it won't be a donut)

matt grime

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if f is an iso from X to Y then fxf is an iso from XxX to YxY sending (a,b) to (f(a),f(b))

in this case yuo can be even more specific since the image of R under arctan is the open interval (-pi/2,pi/2), and you know what that maps to under exp, right?

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