(adsbygoogle = window.adsbygoogle || []).push({}); Need help in some "inter-dimensional isomorphisms"

consider the set

M = {e^(i*arctan(x)) inC| x inR}

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 toR^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 :) inR^4 orC^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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# Need help in some inter-dimensional isomorphisms

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