1. Compactness in Metric space

2. Compact X: metric space X is compact if and only if any sequence in
X has a subsequence which converges in X.

3. Observation
A compact metric space is complete.

5. Observation (X, ρ): metric space is compact if (K, ρ) is compact.
then K is closed and bounded.
closed and bounded
In Rn, compact
see next page
But It does not hold in ℓ2

7. Totally bounded A metric space X is called totally bounded ,there are
if for any
such that
The set
is called an ε-net of X.

8. Observation
(X, ρ): metric space
boundedness
Totally bounded

9. Theorem If X is a complete metric space, then
X is compact if and only if
X is totally bounded.

13. Corollary Let X be a metric space.
If X is compact, then X is separable.

15. Corollary Let X be a compact metric space.
The topology of X is generated by a
countable number of balls of X.

16. Theorem A metric space X is compact if and only if every open cover
of X contains a finite subcover of X.

20. C(S) S: compact metric space
C(S): the space of all continuous real-valued
functions defined on S.
For
f is uniformly continuous

21. Uniformly Bounded M is called an uniformly bounded family if
M is bounded in C(S)

22. Equicontinuous M is called equicontinuous if for any ε>0,
with d(s,t)<δ
there is a δ>0 s.t.

23. Theorem (Arzelá-Ascoli)
is compact
A closed set
if and only if
(i) M is bounded on C(S)
(ii) M is equicontinuous