Closedness in topological product

**Amanda1990** · February 24th 2009, 08:18 AM

Let $X$ and $Y$ be topological spaces with $Y$ compact and let $p_X : X \times Y \rightarrow X$ denote the projection of the topological product on $X$ , $p_X (x,y) = x$ . Given that $W \subseteq X \times Y$ is closed in $X \times Y$ , prove that $p_X (W)$ is closed in X.

**aliceinwonderland** · February 24th 2009, 06:58 PM

Originally Posted by Amanda1990

Let $X$ and $Y$ be topological spaces with $Y$ compact and let $p_X : X \times Y \rightarrow X$ denote the projection of the topological product on $X$ , $p_X (x,y) = x$ . Given that $W \subseteq X \times Y$ is 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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