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16228654 No.16228654 [Reply] [Original]

Is R^2 isomorphic to C?

>> No.16228659

Only topologically.

>> No.16229412

Define "isomorphic". But no.

>> No.16229518

>>16228654
yes if you define multiplication appropriately

>> No.16229746

>>16228659
do you mean they are homeomorphic?

>> No.16229749

>>16229746
they must be since there's a bijection R^2 -> C and (x,y) -> x + iy

>> No.16229909

>>16228654
They're isomorphic as normed vector spaces over the reals. C is also a ring and a vector space over itself, and R^2 isn't, which makes them not isomorphic in those categories.