1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: R^2 homeomorphic to R^n

  1. Nov 9, 2008 #1


    User Avatar

    1. The problem statement, all variables and given/known data

    Prove that [tex]R^2[/tex] and [tex]R^n[/tex] are not homeomorphic if [tex]n\neq2[/tex] (Hint: Consider the complement of a point in [tex]R^2[/tex] or [tex]R^n[/tex]).

    2. Relevant equations

    3. The attempt at a solution

    The proof that [tex]R^n[/tex] is not homeomorphic to [tex]R[/tex] is done by considering that if they are homeomorphic i.e. there exists a continuous bijection with continuous inverse [tex]f:R\rightarrow R^n[/tex]. Then the restriction [tex]f:R\backslash {0} \rightarrow R^n\backslash f(0)[/tex] is also a continuous bijection. Since [tex]R\backslash {0}[/tex] is not connected but [tex]R^n\backslash f(0)[/tex] is if [tex]n\neq 1[/tex] we have a contradiction.

    However, to show that [tex]R^2[/tex] is not homeomorphic to [tex]R^n[/tex] this doesn't work. There is also no other topological invariant I could detect. Both [tex]R^2[/tex] and [tex]R^n[/tex] are contractible to a point and thus have the same fundamental group, for example.

    I would appreciate any idea
  2. jcsd
  3. Nov 9, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Instead of taking out a point, try taking out something bigger :)
  4. Nov 9, 2008 #3


    User Avatar
    Science Advisor

    You can continue that idea. In R2\{0}, a closed loop containing a point p cannot be contracted to a point. In Rn, for n larger than 2, it can.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook