Equivalent Definitions of Compactness

Theorem

The following statements are equivalent:

1. is compact.

2. For every familyof closed subsets ofcontains a finite subclasssuch that

Proof

Supposethen from De Morgans's Laws,

is an open cover ofbecause all theare closed.

Sinceis compact a finite subcoverexists.

Again from De Morgans' Laws,

Conversely, letbe an open cover ofso thatwhere eachis open in

From De Morgans' Laws,

All theare closed and have empty intersection. A subclass ofexists such that

Again using De Morgan's Laws,andis compact.