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Problem: Let

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

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

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

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

[tex] \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\}.[/tex]

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

Any hints?

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# Quotient Spaces/Homeomorphic spaces?

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