# Quotient Spaces/Homeomorphic spaces?

1. Apr 29, 2010

### 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?

Last edited: Apr 29, 2010
2. Apr 29, 2010

### 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.