Definition – Compact set. – Math-reference.net

Definition

Let $F$ be a family of open sets and let $A$ be a set. $F$ is compact if its containment in the union of all the sets in $F$ implies that it is contained in some finite number of the sets in $F$ .

Notes

Related to this definition are:
1. Open cover.
2. (Open) subcover.
3. Heine-Borel theorem.

First of all, it is important to not develop an intuition which goes from a family of sets to the set $A$ , but rather to begin with a set $A$ and arrive at a family of open sets to which the union operator could be applied to form $A$ .

For example, rather than considering a family of open sets $F$ , and postulating for which sets $F$ is an open cover, it may be of more use to consider a specific set $A$ , for example the interval $A = [0,~1]$ , for which we need to find a family of sets $F$ which “cover” $A$ . In this case, the family $F = \{(-1,2), (2, 3)\}$ would be an open cover of $A$ , since the first interval in the set contains $A$ .