Definition A set E is compact if and only if, for every family of open sets such that , there is a finite set such that .

Example: Let E=(0,1] and for each positive integer n, let . If , there is a positive integer n such that ; hence, , and thus

If we choose a finite set of positive integers, then

where and

Thus, we have a family of open sets such that , but no finite subfamily has this property. From the definition, it is clear that E is not compact.

Heine-Borel Theorom: A set is compact iff is closed and bounded.

Examples:

Note that all of these examples are of sets that are uncountably infinite.

Definitions from: Introduction to Analysis 5th edition by Edward D. Gaughan

Theorom: The union of compact sets is compact.

CompactSet (last edited 2020-01-26 17:51:19 by scot)