- Thread starter
- #1

dimensional sphere S^2 where two points are equivalent if and only if they

are equal or antipodal, i.e.,

x ~ y , ⇔ x = +y or -y:

Let X be the set of equivalence classes S^2/~

with the quotient topology inherit from S2

Consider the equivalence relation on the closed

2-disk D^2 where two points are equivalent if and only if either they are equal or they are antipodal

points on ∂D^2

i.e.,

x ~ y ⇔ (x = y) or (x,y ∈∂D^2 and x=y or -y)

Let Y be the set of equivalence classes D^2/~ with the quotient topology inherit from D^2

.

Prove that X and Y are homemorphic

I was trying to find a bijective function between X and Y and to prove both the function and its inverse are continuous, However, I did not find the function, is anyone can help.

thanks