Let $(X, T_1)$ and $(Y,T_2)$ be topological spaces. Prove that if $(X\times Y,T)$ is compact, then each $(X,T_1)$, $(Y,T_2)$ is compact.
Here is how I start this problem: Since $(X\times Y, T)$ is compact, each of its open covers has a finite subcover $O_{x_1}\times O_{y_1}$,…,$O_{x_n} \times O_{y_n} $
So, there is a finite number of $O_{x_1},…,O_{x_n}$. This is a finite subcover for X.
and there is a finite number of $O_{y_1},…O_{y_n}$. This is a finite subcover for Y
Therefore, $X$ and $Y$ are each compact. Please check my proof for me?
Thanks
Best Answer
You’re off on the wrong foot altogether. If you’re going to work directly with open covers (instead of @lhf’s idea), you must start with an open cover of $X$ (or $Y$) and show that it has a finite subcover. Starting with an open cover of $X\times Y$ is pointless: you’re not trying to prove that $X\times Y$ is compact. (And note that open sets in $X\times Y$ do not necessarily have the form $U\times V$, where $U$ is open in $X$ and $V$ is open in $Y$.)
Let $\mathscr{U}$ be an open cover of $X$. For each $U\in\mathscr{U}$ let $V_U=U\times Y$, and let $\mathscr{V}=\{V_U:U\in\mathscr{U}\}$. Show that $\mathscr{V}$ is an open cover of $X\times Y$ and apply compactness of $X\times Y$ to get a finite subcover of $\mathscr{V}$. Use this finite subcover to find a finite subset of $\mathscr{U}$ that covers $X$.
The proof that $Y$ is compact is entirely similar.