Page 77 - DMTH401_REAL ANALYSIS

Unit 4: Compactness

Notes
æ r(x ) æ r(x )
j ö
jk ö
B x , j ÷ for i > j. This has convergent subsequence x by assumption, so  k s.t. B x , .
ç
è 2 ø jk ç è jk 2 ø ÷
This is a contradiction, so the procedure stops.
Self Assessment
Fill in the blanks:

1. T, S compact, U open cover of ............... If s  S there exists open V  S, s  V s.t. T  V can
be covered by finite subfamily of U.
2. Let T be compact. Cts ............................ is bdd and attains max and min.

3. Given cover U of metric M,  > 0 called ...................... of U if " x M U U s.t. B(x, )  U.

4.8 Summary

 Let F F ... sequence of non-empty closed subsets of compact T. Then  ¥ F   . Let F
=
1 2 k 1 k 1
 F ... sequence of non-empty compact subsets of Hausdorff T. Then  ¥ F   .
=
2 k 1 k
 " x  T find W  U s.t. (x, s) W . Exists open U  T, V  S s.t. (x, s)  U V  W .
x
x
x
x
x
x
x
{U : x  T} open cover of T so U ,..., U which cover T. Let V =  n i 1 V . V  S open and
x x1 xn = xi
n n
T  V   U  V xi  W xi
xi
i 1 i 1
=
=
 Let x   ¥ j 1 {x , x ...}.  k  j s.t. x  x. k ¥ so can choose subsequence k s.t. k > k ji
=
j
+
j 1
ji
kj
j
ji+1
j
(as ks not necessarily in order). Then x subsequence converging to x.
j kji
 Let x  M, F = {x , x ...}. F form decreasing sequence of non-empty closed subsets of M.
+
k j j j 1 j
 ¥ j 1 F   so x have convergent subsequence.
=
k
j
4.9 Keywords
Space Compact: Cts image of compact space compact.
Homeomorphism: Cts bijection of compact T onto Hausdorff S is homeomorphism.
Lebesgue Number: Every open cover U of compact metric space has a Lebesgue number.
Convergent Subsequence: Metric M is compact iff every sequence in M has convergent subsequence.
4.10 Review Questions
1. Discuss the compactness of a set.
2. Explain intersection of closed set.
3. Discuss compactness and Continuity.
4. Describe sequential compactness.

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