[Cardinality] Prove there is no bijection between two sets

[mick25]

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

?

 

[fauboca]

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

?

How about something easier?

Is R compact? Is the unit circle compact?

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

 

[mick25]

How about something easier?

Is R compact? Is the unit circle compact?

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

nope

i just realized it after posting this thread but i don't know how to delete it now

thanks