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Quotient topology

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New member
Oct 15, 2013
19
Consider the equivalent relation on the two
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