Page 237 - DMTH503_TOPOLOGY

Unit 28: Compactness in Metric Spaces

Suppose the contrary. Notes
Then  no Lebesgue number for the cover {G } . Then for each n N,  a set B  X with the
i i e D n
1
property that 0 < d (B ) <
n n
and B  G  i   
n i
Choose a point b  B  n  N and consider the sequence <b >. By the assumption of sequential
n n n
compactness, the sequence <b : n  N> contains a subsequence <b : n  N> which converges to
n in
b  X.
But {G } is an open cover of X so that
i
 open set G s.t. b  G . By definition of open set
i i
0 0
S  G ...(2)
(b) i
0
b  b
i
n

 Given any  > 0,  n  N s.t.  i  n  b  S (b). ...(3)
0 n 0 i
n 2
Choosing a positive integer K ( n ) such that
0 0
1 
 ...(4)
K 2
0
From (3), i  K  b  S (b)
n 0 n i  /2
 In particular bK  S (b) ...(5)
0 /2
In accordance with (1)
1
b  B , 0  d(B )  ...(6)
K k K
0 0 0 K
0
On using (4)
0 < d(B ) < /2 ...(7)
K 0
From (5) and (6), if follows that

B S (b)   ...(8)
K 0 /2
 
From (7) and (8), if follows that B is a set of diameter  and it intersects S .2.(b), Showing
K 0 2 2
thereby


B  S .2.(b)
2
K0
i.e., B  S (b).
K0 
In view of (2), this gives B  G ...(9)
K i
0 0
In accordance with (1), B  G , i  
K i 0
0 0
In particular, B  G , i  
K i 0
0 0
Contrary to (9).
Hence the required results follows.

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