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Thread: Projection map

  1. #1
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    Projection map

    I am stuck by this question,

    "In general show that for all $\displaystyle j \in I$ there is a map

    $\displaystyle \pi_j : \prod_{i \in I} X_i \rightarrow X_j$

    which is the 'projection onto the j-th factor' ".


    I'm stuck because I didn't even know you could ask this question; I'm all at sea about how to even do this!

    Note, that $\displaystyle \pi_j : A_1 \times A_2 \times \cdots \times A_j \times \cdots \times A_n \rightarrow A_j$, given by $\displaystyle \pi_j(a_1, a_2, \ldots, a_j, \ldots, a_n) = a_j$
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by HTale View Post
    I am stuck by this question,

    "In general show that for all $\displaystyle j \in I$ there is a map

    $\displaystyle \pi_j : \prod_{i \in I} X_i \rightarrow X_j$

    which is the 'projection onto the j-th factor' ".


    I'm stuck because I didn't even know you could ask this question; I'm all at sea about how to even do this!

    Note, that $\displaystyle \pi_j : A_1 \times A_2 \times \cdots \times A_j \times \cdots \times A_n \rightarrow A_j$, given by $\displaystyle \pi_j(a_1, a_2, \ldots, a_j, \ldots, a_n) = a_j$
    I'm not sure I grabbed the problem correctly...

    Let's assume $\displaystyle I=\{i_1,i_2,\dots,i_j,\dots \}$ (a countable set, otherwise it makes no sense to write the product)
    Let $\displaystyle x \in \prod_{i \in I} X_i$.
    Then x can be written as a tuple :
    $\displaystyle x=(x_{i_1},x_{i_2},x_{i_3},\dots,x_{i_j},\dots)$, where $\displaystyle x_{i_k} \in X_{i_k}$ for any $\displaystyle i_k \in I$

    $\displaystyle n \in I$ means that there exists $\displaystyle j$ such that $\displaystyle n=i_j$

    And $\displaystyle \pi_n$ will yield the $\displaystyle i_j$-th coordinate of x, that is $\displaystyle x_{i_j}$

    So it's defined... ?
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