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Homework Help: Quotient map on a product space

  1. Jan 10, 2007 #1
    suppose q:M -> M/R is a quotient map.

    i've asked my dad what is the quotient map from MxM to (M/R)x(M/R)?

    he told me it is qxq: MxM -> (M/R)x(M/R) defined by

    (qxq)(x,y) = (q(x), q(y)),

    but there are some conditions to be met, but he could not remember what those conditions are. i went through all 6 or so of my topology textbooks and could not find it.

    does anybody know what the conditions are for qxq: MxM -> (M/R)x(M/R) defined by

    (qxq)(x,y) = (q(x), q(y)),

    to be a quotient map if q:M -> M/R is a quotient map???
     
  2. jcsd
  3. Jan 13, 2007 #2

    Gib Z

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    Homework Helper

    It seems like no one wants to answer your question :S

    In fact, the only reason Im posting is so this thread goes to the top of the list so someone might notice it, because I sure as hell don't know it.

    Good Luck
     
  4. Jan 15, 2007 #3
    thanks a lot gib, i accepted already that apparently no one knows. the munkres topology book gives a counterexample of where a product of quotient maps is not a quotient map and i can see why it fails in the counterexample but can't figure out the general condition for failure. the counterexample munkres gives is so weird that i assume that qxq: MxM -> (M/R)x(M/R) is a quotient map in most cases except for the bizzarre ones.

    thanks buddy for watching out for me. i took your advice about learning math for the sake of discovering what i really like, rather than just trying to scale up mount everest as quickly as possible. in fact, today i reverted to rereading the proofs to basic set theory to make sure i master the basics before jumping to 4th year courses.
     
  5. Jan 16, 2007 #4

    Hurkyl

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    Gold Member

    Well, we can look at the pieces.

    As a function on the underlying sets, qxq is certainly a surjective function, and "is" thus a quotient map.

    As a map between topolocial spaces, qxq is clearly a continuous function.

    So, the only possibility is that the topology on (M/R)x(M/R) is not the one induced by the map qxq, right?

    ...
     
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