# Homework Help: [Cardinality] Prove there is no bijection between two sets

1. Feb 27, 2012

### mick25

1. The problem statement, all variables and given/known data
prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R

2. Relevant equations

3. 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

???

2. Feb 27, 2012

### fauboca

Is R compact? Is the unit circle compact?

Can you have continuous map from a compact set to a non compact set?

3. Feb 27, 2012

### mick25

nope

i just realized it after posting this thread but i dont know how to delete it now

thanks