Show that if \(A\) is closed in \(X\) and \(B\) is closed in \(Y\), then \(A \times B\) is closed in \(X \times Y\).
Solution
Since \(A\) and \(B\) are closed, \(X-A\) and \(Y-B\) are open. The complement of \(A\times B\) is \[ (X\times Y)-(A\times B) =((X-A)\times Y)\cup(X\times(Y-B)), \] a union of product-open sets. Therefore \(A\times B\) is closed in \(X\times Y\).