Thread: Closedness in topological product

1. Closedness in topological product

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.

2. 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 $X$.