Page 74 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Then a A so A , bounded above by b. Let c = sup A. a  c  b so c U for some U U. U open
so  > 0 s.t. (c – , c + ) U.
c = sup A so  x  A s.t. x > c – . [a, c + )  [a, x]  (c – , c + ) can be covered by finite subfamily
of U so (c, c +)  [a, b] =  (since any point in here is in A but > c sup A). So c = b.

4.2 Compactness of Subsets

Proposition: Any closed subset C of compact space compact.

Proof: Let U be cover of C by sets open in T. Adding open T\C get open cover of T. Finite subcover
of this cover contains finite subcover of C of sets from U.
Proposition: Compact subspace C of Hausdorff T is closed in T.

Proof: a T\C." x  C  disjoint U ' x, V ' a open in T since T Hausdorff. U open cover of C so
x x x
n
has finite subcover U ,... , U . Then V =  i 1 V open, a  V and disjoint from C. Hence
xl xn = xi
a  (T\C)° and T\C open.
Proposition: Compact subspace C of metric space M is bounded.
Proof: Let a  M. Balls B(a, r) (r > 0) are open and cover C, so  r ,... , r s.t. C   n i 1 B(a, r ) =
1 n = i
B (a, max {r , .... , r }).
1 n
4.3 Intersections of Closed Sets

Theorem: Let F be collection of non-empty closed subsets of compact T s.t. every finite subcollection
of F has non-empty intersection. Then intersection of all sets from T non-empty.
Proof: Assume intersection of all sets empty. Let U be collection of complements. U covers T by
DeMorgan. U open cover so exists finite subcover U ,..., U . Then F : = T\U  F and empty
1 n i i
intersection by DeMorgan. This contradicts the assumption of the theorem.
Corollary: Let F F ... sequence of non-empty closed subsets of compact T. Then  ¥ F   .
=
1 2 k 1 k
Corollary: Let F  F  ... sequence of non-empty compact subsets of Hausdorff T.
1 2
Then  ¥ F   .
=
k 1 k
Proof: By proposition 4.4 compact subsets of Hausdorff space are closed.
4.4 Compactness of Products

Lemma: T, S compact, U open cover of T S. If s  S there exists open V  S, s  V s.t. T  V can
be covered by finite subfamily of U.
Proof: " 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
=
Theorem: (Tychonov). S,T compact  T  S compact.
Proof: By lemma 3.8 " y  S  V S open s.t. T  V can be covered by finite subfamily of U. S
y y
compact, {V : y  S} form open cover so  V ,..., V which cover S.
y yi ym

68 LOVELY PROFESSIONAL UNIVERSITY