Proof That The Cartesian Product of Two Closed Sets is a Closed Set

Theorem

The cartesian product of two closed sets is a closed set.

Proof

Letandbe topological spaces and letandbe closed.

The set (X times Y) - (A times B) is the union of two open sets, so is an open set.

Similarly ifandare T2 spaces then the cartesian product is a T2 space. Regularity is also preserved by the Cartesian product. Normality however is not preserved.