- #1
mick25
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Homework Statement
prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R
Homework Equations
The Attempt at a Solution
is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between those two sets
R has the cardinality of continuum
the unit circle is defined on [-1,1]x[-1,1] and since [a,b] has same cardinality as R for all a,b, cardinality of the unit circle would be c*c = c^2 but c^2=c, but this can't be since then there would be a bijection between the unit circle and R
?