Equivalent Definitions of Compactness
The following statements are equivalent:
For every familyof closed subsets ofcontains a finite subclasssuch that
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.