- Thread starter
- #1

#### Julio

##### Member

- Feb 14, 2014

- 71

$A=X\cap F_1$ where $F_1$ is closed. Furthermore, $B=Y\cap F_2$ where $F_2$ is closed. It follows that $X$ and $Y$ are closed. Thus,

$

\begin{eqnarray*}

A\times B &=& (X\cap F_1)\times (Y\cap F_2)\\

&=& (X\times Y)\cap (F_1\times F_2)\\

&=& \text{closed in}\, X\times Y

\end{eqnarray*}

$

Finally, at $X$ and $Y$ are closed, then $X\times Y$ is closed. Analogous, $F_1\times F_2$ is closed, because $F_1,F_2$ are closed.