Quotient Spaces/Homeomorphic spaces?

[sutupidmath]

Hi,

Problem: Let

X=\{x\times y|x^2+y^2\leq1\}, \mbox{ in } R^2.

\mbox{ Let } X^{\star} \mbox{ be the partition of X consisting of all the one point sets } \{x\times y\},

x^2+y^2<1, \mbox{ along with the set } S^1=\{x\times y | x^2+y^2=1\}.

\mbox{ Then it continues by saying that one can show that } X^{\star}

\mbox{ is homeomorphic with the subspace of } R^3 \mbox { called the unit 2-sphere, defined by } S^2=\{(x,y,z)|x^2+y^2+z^2=1\}.

My question is how would one build a homeomorphism between these two spaces?

Any hints?

 

[Bacle]

Think of what space you get when you remove one point from the sphere, and how it
relates to X/~ without the boundary points class.