- #1
alexbennie
- 1
- 0
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
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