Page 242 - DMTH503_TOPOLOGY

Topology

Notes 
<  
2
1 1

 d p, a n  d a , b  n n  <     
0
0
2 2
0
<  +  = d (p, B)
d
 p, b n  since b  B.
n
0 0
or d p, a n 0  d a , b  n 0 n 0  d p, b  n 0 
This contradicts the triangle inequality.
Thus d(p, B) = d(A, B).

28.3 Summary

 A closed and bounded infinite subset of R contains a limit point.
 A metric space (X, d) is said to have the BWP if every infinite subset of X has a limit point.

 A metric space (X, d) is said to sequentially compact if every sequence in X has a convergent
subsequence.
 Let {G : i  } be an open cover for a metric space (X, d). A real number  > 0 is called a
i
Lebesgue number for the cover if any A  X s.t. d(A) < d  A  G for at least one index
0 i
i  .
0
 Every open covering of a sequentially compact space has a lebesgue number.

 If (X, d) be a metric space and A  X, then the statement that A is compact, A is countably
compact and A is sequentially compact are equivalent.

28.4 Keywords

Cauchy sequence: Let <x > be a sequence in a metric space (X, d). Then <x > is called a cauchy
n n
sequence if given  > 0,  n  N s.t. m, n  n  d(x , x ) < .
0 0 m n
Compact: Let (X, T) be a topological space and A  X. A is said to be a compact set if every open
covering of A is reducible to finite sub covering.
Complete metric space: Let (X, d) a metric space then (X, d) is complete if cauchy sequence of
elements of X converges to some elements (belonging to X).

Equicontinuous: A collection of real valued functions.
A = {f : f : X  R} defined on a metric space (X, d) is said to be equicontinuous if
n n
given  > 0,   = () > 0 s.t.

d(x , x ) <   | f(x ) – f(x ) | <   f A.
0 1 0 1
Finite subcover: If  G  G s.t. G is a finite set and that {G : G  G } is a cover of A, then G is
1 1 1 1
called a finite subcover of the original cover.
Open cover: If every member of G is an open set, then the cover G is called an open cover.

236 LOVELY PROFESSIONAL UNIVERSITY