Proof That The Cartesian Product of Two Closed Sets is a Closed Set
The cartesian product of two closed sets is a closed set.
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.