MHB Projection Map $X \times Y$: Closure Property

[Euge]

Here is this week's POTW:

-----
Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed.
-----

 

[Euge]

No one solved this problem. You can read my solution below.

Spoiler

Let $C$ be a closed set in $X \times Y$. If $x\in X\setminus p_X(C)$, then for every $y\in Y$ the ordered pair $(x,y)\notin C$; it follows that there are open neighborhoods $U_y$ of $x$ and $V_y$ of $y$ such that $(U_y\times V_y)\cap C = \emptyset$. The collection $\{V_y:y\in Y\}$ is an open cover of $Y$; by compactness of $Y$, there are $y_1,\ldots, y_n\in Y$ such that $Y = V_{y_1}\cap \cdots \cap V_{y_n}$. Let $U = U_{y_1}\cap \cdots \cap U_{y_n}$. Then $U$ is an open neighborhood of $x$ such that $(U\times Y) \cap C = \emptyset$, i.e., $p_X^{-1}(U)\cap C = \emptyset$. Thus $U \cap p_X(C) = \emptyset$. Since $x$ was arbitrary, $p_X(C)$ is closed.