5.2. Compact and Perfect Sets

Examples 5.2.5(c):

Let S = (0, 1). Define a collection C = { (1/j, 1), for all j > 0 }. Is C an open cover for S ? How many sets from the collection C are actually needed to cover S ?
Clearly, each set in the collection C is open. Also so that C is indeed an open cover of S.

Not all sets from the original collection C are needed to cover S. For example, the subcollection of intervals (1 / (2 j) , 1) is also an open covering. However, we can not reduce this cover to a finite subcovering. To see this, extract finitely many sets of the form (1 / j , 1) from the collection C. Let N be the largest integer j that occurs in this subcollection. Then the point 1 / (N + 1) is in S, but it is not in any of the intervals of the finite subcollection. Hence, no finite subcollection from C can cover S.

On the other hand, S does have some other finite open coverings. For example, the collection { (-1, 1/2), (0, 2) } is such a finite open cover. However, the giving open covering C from above can not be reduced to a finite subcover.

Next | Previous | Glossary | Map