We claim that no point has a compact neighbourhood. The topological properties possessed by a product depends, of course, on the properties possessed by the individual factors. There are several theorems which assert that certain topological properties are productive i.e. are possessed by the product if enjoyed by each factor. Several of these theorems are given below. Any product of connected spaces must be connected.
Proof is left to the reader.
Any product of compact spaces is compact i.e. compactness is productive. It suffices to prove that any covering of X by open cylinders has a finite subcover. Suppose not and let C be a family of open cylinders which covers X but for which no finite subcover exists. Since C covered X, z some C C . To prove the above without Alexander's Subbase Theorem is very difficult in general, but it is fairly simple in the special case where I is finite.