Results 1 to 2 of 2

Math Help - Topology question: continuous maps and homeomorphisms

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    36

    Topology question: continuous maps and homeomorphisms

    Let R denote the real numbers.
    Let S^2 denote the unit sphere in R^3
    Let f: S^2 -> R^4 be defined by f(x,y,z)=(x^2-y^2,xy,yz,zx)

    Can we prove that:

    f determines a continuous map g: PR^2 -> R^4 where PR^2 is the real projective plane.
    &
    g is a homeomorphism onto a topological subspace of R^4
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23
    for the first question we need only to verify that f(p)=f(-p), which is obvious. So if \pi is the quotient map from S^2 to RP^2, q= \pi(p)= \pi(-p) is the image of p, g(q) is defined to be f(p). g is obviously continuous since g=f(\pi^{-1}), if we choose a sheet of the covering \pi locally.
    To see that g is a homeomorphism from M= RP^2 to g(M), you need only to show that g is injective.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. different notions of continuous maps
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 9th 2011, 05:30 AM
  2. Topology homeomorphisms and quotient map question
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 22nd 2011, 12:24 PM
  3. Continuous functions and homeomorphisms
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 15th 2009, 06:51 AM
  4. Replies: 1
    Last Post: October 20th 2008, 02:44 PM
  5. [SOLVED] Continuous Functions/Maps
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 23rd 2007, 04:30 AM

Search Tags


/mathhelpforum @mathhelpforum