120 Chapter 2
EXERCISES 2. 7
1. If (S, d) and (T, d') are complete metric spaces, show that the product
space S x T with metric
p[(x1,Y1),(x2,Y2)] = y'd^2 (x1,x2) +d'^2 (Y1,Y2)
is complete. In particular, it follows that a Euclidean space of dimension
n ~ 2 is complete.
- Show that the space C[a, b] of all continuous functions x(t) on [a, b] with
metric
is complete.
d(x,y) = max lx(t)-y(t)I
a::;t:::;b
- Let (S', d) be a metric space. Prove the following.
(a) If M is a subspace of S( with the induced metric) and M is complete,
then M is closed.
(b) If S is complete and M C S is closed, then M is complete.
- Show that (<C*, x) is complete.
- Prove that a metric space is compact iff it is both complete and totally
bounded.
2.13 TOPOLOGICAL SPACES
Certain important properties of metric spaces (e.g., that of compactness)
depend only of the notion of open set (or, equivalently, on the notion of
closed set). The concept of an open set depended, in turn, on the metric
employed. In some later work it proves convenient to free certain notions
(proximity, continuity) from that of a distance. With this purpose we
discuss in this section the concept of a topological space, which is based on
an abstraction of the concept of open set.
Definition 2.46 A topological space (S,D) is a nonempty set S of ele-
ments of an arbitrary nature (called the points of the space) together with
a collection n of subsets of S (called the open sets) having the following
properties:
- 0 E D and S E D
2. If A; E D ( i E I), then LJiEI Ai E D (the union of any collection of
open sets is open)
- If A; E D ( i = 1, 2, ... , n) then n~=l Ai E n (the intersection of a finite
collection of open sets is open).
