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: Topology Problem

  1. Apr 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Find two spaces A,B compact where A and B are nonhomeomorphic but AX[0,1][tex]\cong[/tex]BX[0,1]

    2. Relevant equations

    Definitions of homeomorphism, cardinality possiby, I have no idea where to start.

    3. The attempt at a solution
    My idea Is [0,1] and S^1, but im not sure if the unit square is homeo to the cylinder.
     
  2. jcsd
  3. Apr 10, 2010 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    It's not. The cylinder has a loop in it that can't be shrunk to a point (one going around it) and the unit square doesn't have anything like that.

    A hint: If f(x,t) is the homeomorphism from Ax[0,1] to Bx[0,1], it must be that if f(x,t)=(y,s) that t is not equal to s in general, otherwise restricting yourself to a single value of t would give a homeomorphism from A to B
     
  4. Apr 10, 2010 #3
    That's actually a really useful idea. For some reason I look at your hint and think homotopy. Is that a step in the right direction?
     
  5. Apr 11, 2010 #4
    How about A= space between two concentric circles, with a smaller circle glued to the inside of the smaller circle, and a circle of the same size glued to the outside, and B= the same concentric circle space, but two circles are on the outside?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook