3.3 * Compact Sets 155
Notice that the following are not topological terms: "interval,'' "bounded,''
"sup A,'' "inf A,'' "Archimedean,'' and "complete.""Compact" is another topological term, although you cannot tell that from
the definition we gave above. It turns out that the notion of a "compact set" is
a very powerful tool in analysis, but its topological definition is rather compli-
cated. After developing the abstract machinery necessary to give a topological
definition, we shall give that definition and develop some consequences of it.
Compact sets share some essential features with finite sets, which make
them especially well-suited for expressing some of the results of real analysis.
In fact, we shall see that all finite sets are compact.
Definition 3.3.2 Suppose A is a set of real numbers.
(a) A family U of open sets of real numbers is said to be an open cover of
A if every element of A belongs to at least one set in U ; that is,
Va E A, 3 U E U 3 a E U;
i.e., A~ u U.(b) If U is an open cover of A and there is a finite subcollection of U that
covers A [i.e., 3 U 1 , U 2 , · · · , Un EU 3 A~ kQl Uk], then we say that U
has a finite subcover of A.Example 3.3.3 Let A= [O, 1] and U = {Ni (r) : r E Q, n EN}·
(a) Then U consists of open intervals ( r - ~, r + ~) that collectively "cover"
[O, l].(^0) r-k r r+ k
Figure 3.9
Thus, U is an open cover of A. Notice that U contains infinitely many sets, so
U is not a finite open cover of A.
(b) We shall now produce a finite subcollection of U that covers A. (Hence,
U has a finite subcover of A.)