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Thread: Closedness in topological product

  1. #1
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    Closedness in topological product

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

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

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

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

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