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Math Help - Closedness in topological product

  1. #1
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    Closedness in topological product

    Let X and Ybe topological spaces with Ycompact and let p_X : X \times Y \rightarrow Xdenote the projection of the topological product on X, p_X (x,y) = x. Given that W \subseteq X \times Yis closed in X \times Y, prove that p_X (W) is closed in X.
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  2. #2
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    Quote Originally Posted by Amanda1990 View Post
    Let X and Ybe topological spaces with Ycompact and let p_X : X \times Y \rightarrow Xdenote the projection of the topological product on X, p_X (x,y) = x. Given that W \subseteq X \times Yis closed in X \times Y, prove that p_X (W) is closed in X.
    Tube lemma: Consider the product space X \times Y, where Y is compact. If N is an open set containing the slice x_0 \times Y of X \times Y, then N contains some tube V \times Y about x_0 \times Y, where V is a neighborhood of x_0 in X.

    We shall show that X \setminus p_X(W) is open in X using a tube lemma, which is equivalent to showing that p_X(W) is closed in X.

    Let x_0 be any element in X \setminus p_X(W). By the tube lemma, there exists a tube V \times Y such that

    x_0 \times Y \subset V \times Y \subset (X\setminus p_X(W))\times Y.

    Now, for any x_0 in X \setminus p_X(W), we have an open set V containing x_0 such that V \subset (X \setminus p_X(W)). Thus, X \setminus p_X(W) is open in X.
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