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Thread: Projections

  1. #1
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    Projections

    Hi guys,

    The question I'm doing states,

    "Let $\displaystyle i_1, i_2, \ldots, i_n$ be indices in $\displaystyle I$, and assume that for each $\displaystyle i \in I$, we are given $\displaystyle U_i \subseteq X_i$. Show that $\displaystyle \bigcap_{1\leq r \leq n} \pi^{-1}_{i_r}(U_{i_r})$ can be described as a product of subsets of $\displaystyle X_i$'s"

    I just want to see if my thinking here is correct.

    We have that $\displaystyle \pi^{-1}_{i} $ is given by the projection onto the ith factor. So, $\displaystyle \pi^{-1}_{i_1} : U_{i_1} \times U_{i_2} \times \cdots \times U_{i_n} \rightarrow U_{i_1}$, for instance. We have the following fact for some $\displaystyle U_1\subseteq X_1, U_2 \subseteq X_2$,

    $\displaystyle \pi^{-1}_1(U_1) = U_1 \times X_2$

    where $\displaystyle \pi_1 : U_1 \times U_2 \rightarrow U_1$

    Now, I know I have to use this to prove the above question, I'm just stuck on how I would represent $\displaystyle \pi^{-1}_{i_r}(U_{i_r})$ using the above, and how to then manipulate the intersections. There is a further property that

    $\displaystyle \pi^{-1}_1(U_1) \cap \pi^{-1}_1(U_2) = U_1 \times U_2$,

    which perhaps may come in useful, although I haven't seen any use for it.

    Thanks in advance,

    HTale.
    Last edited by HTale; Feb 8th 2009 at 04:07 PM.
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  2. #2
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    Quote Originally Posted by HTale View Post
    $\displaystyle \pi^{-1}_{i_1} : U_{i_1} \times U_{i_2} \times \cdots \times U_{i_n} \rightarrow U_{i_1}$, for instance. We have the following fact for some $\displaystyle U_1\subseteq X_1, U_2 \subseteq X_2$

    It should be
    $\displaystyle \pi_{i_1} : U_{i_1} \times U_{i_2} \times \cdots \times U_{i_n} \rightarrow U_{i_1}$.

    Let S be a subbasis for the product topology you mentioned. Then,

    $\displaystyle S = \{\pi^{-1}_{i}(U_{i}) : U_{i}$ is open in $\displaystyle X_{i}, i \in I \}$.

    Since the all finite intersection of elements of S forms a basis for (X, T),
    $\displaystyle \bigcap_{1\leq r \leq n} \pi^{-1}_{i_r}(U_{i_r})$ is a basic open set for a topology T. If we denote a basic open set for a product topology T using a product form,

    $\displaystyle \bigcap_{1\leq r \leq n} \pi^{-1}_{i_r}(U_{i_r}) = \prod _{i \in I} A_{i}$, where $\displaystyle A_{i} = U_{i}$ for $\displaystyle i = i_1, i_2, .., i_n $ in $\displaystyle I $ and $\displaystyle A_{i} = X_{i}$ otherwise.
    Last edited by aliceinwonderland; Feb 9th 2009 at 02:32 AM.
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