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Math Help - Product Spaces & Homeomorphism

  1. #1
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    Product Spaces & Homeomorphism

    I have a question that I can't start. I think if I got the start of it I could finish it but I dont know what my first move would be


    Let X,T be the product space \pi(X_i,T_i). Given any 'origin' \bar{x}=(\bar{x_i})_{i+1} in X and any i_0 \in I.

    Show that the map
    \mu: x_i_{0} \rightarrow X given by

    \mu(y) = (x_i)_{i+1} where x_i = \bar{x_i} if i is not equal to i_0 or x_i = yif i=i_0

    is a homeomorphism between x_i_{0} and \mu(x_i_{0})
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  2. #2
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    Quote Originally Posted by Turloughmack View Post
    I have a question that I can't start. I think if I got the start of it I could finish it but I dont know what my first move would be


    Let X,T be the product space \pi(X_i,T_i). Given any 'origin' \bar{x}=(\bar{x_i})_{i+1} in X and any i_0 \in I.

    Show that the map
    \mu: x_i_{0} \rightarrow X given by

    \mu(y) = (x_i)_{i+1} where x_i = \bar{x_i} if i is not equal to i_0 or x_i = yif i=i_0

    is a homeomorphism between x_i_{0} and \mu(x_i_{0})

    Perhaps it's only me, but I can't make any sense of your post: you introduce us to X,T (presumably, X is

    a points set and T is a topology on it), then you write \pi(X_i,T_i) , which I guess (guess!) means the product of

    top. spaces (X_i,T_i) , though the little pi is the usual notation, within this context, for the fundamental group.

    Thus, in fact it seems to be that X=\prod\limits_{i\in I}X_i , whereas T is the heaven knows what topology on this
    (The prod. topology? The box topology? Other topology?) . You then take an "origin" \bar{x}=(\bar{x_i})_{i+1}\in X\,,\,\,i_0\in I ...why

    the index is i+1?? And what or who is i_0? The set I is, I presume, the original one indexing the product above...?

    Then you define a map from x_{i_0} (What is this? A coordinate from the element (x_i) above, or what?) to X , but

    you define it on some y....and then you ask to show that x_{i_0} is homeom. to its image...?

    Please try to be way clearer or, preferably, post a link to the original question...or dismiss this post

    if I made a whole mess from something simple.

    Tonio
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