Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quotient Spaces/Homeomorphic spaces?

  1. Apr 29, 2010 #1

    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?
    Last edited: Apr 29, 2010
  2. jcsd
  3. Apr 29, 2010 #2
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook