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Quotient map (A simple one).

  1. Apr 18, 2008 #1
    Im reviewing material for the exam and came across this question:
    Let pi_1:RxR->R be the projection on the first coordinate.
    Let A be the subspace of RxR consisitng of all points (x,y) s.t either x>=0 or (inclusive or) y=0.

    let q:A->R be obtained by resticting pi_1. show that q is quotient map that is neither open nor closed.
    now to show that it's quotient map is the easy task, I want to see if I grasp it correctly, the set [0,infinity)x{0} is closed and open in A, but q([0,infinity)x{0})=[0,infinity) isn't closed nor open in R, correct?
     
  2. jcsd
  3. Apr 18, 2008 #2

    [0,inf)x{0} is certianly not closed and open because A is simply connected and so A itself and the empty set are the only closed&open sets.

    I guess you will have to find two sets, one closed, the other open, such that their image is not closed (open).
     
  4. Apr 19, 2008 #3
    I understand that [0,infinity) is closed in R, cause it's the complement of (-infinity,0) which is open in R.
    So how to find such examples?
    I mean for example a closed set in A would be an intersection of A with a closed set in RxR, now because q gives us only the first coordinate, then A obviously consists of all the points of the form: [0,infinity)xR and Rx{0}, which means that the map of such sets under q would be: subsets of [0,infinity) or R.
    I don't see any example that shows what i need.
     
  5. Apr 20, 2008 #4

    WWGD

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    RxR, now because q gives us only the first coordinate, then A obviously consists of all the points of the form: [0,infinity)xR and Rx{0}, which means that the map of such sets under q would be: subsets of [0,infinity) or R.
    I don't see any example that shows what i need.[/QUOTE].


    I think the standard example is that of the subset A={ (x,y) in IR^2 : yx=1 }.




    I don't know of a general way of generating (counter) examples, but there is

    a result that quotient maps take saturated open ( equiv. closed) sets to

    open (equiv. closed) sets. So if you can consider subsets that are not saturated

    under your map, this should help.
     
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