
“real: chapter_04” — 2011/5/22 — 23:17 — page 26 — #26
4-26 Real Analysis
The above theorem tells us that those subsets of X , which are com-
plete and totally bounded, share the property of closed and bounded
subsets of
R mentioned in Theorems 4.2.2. In addition, these type
of sets are also closed and bounded in X . On the other hand, as we
have already observed closed sets (in general metric spaces) need not
be complete and bounded sets need not be totally bounded. For this
reason we shall give a special name for these types of sets as follows.
Definition 4.3.20 Let (X , d) be a metric space and S ⊂ X . S is said
to be compact if every open cover of