Page 135 - DMTH503_TOPOLOGY

Unit 13: Compact Spaces and Compact Subspace of Real Line

Theorem 8: Every closed and bounded interval on the real line is compact. Notes
Proof: Let I = [a, b] be a closed and bounded interval on . If possible, let I be not compact. Then
1 1
there exists an open covering  = {G } of I , having no finite sub covering.
i 1
 a b  a  b 
 
Let us write I = [a, b] = a,  , b …(1)
1    
 2   2 
Since I is not covered by a finite sub-class of  and therefore at least one of the intervals of the
1
union in (1) cannot be covered by any finite sub-class of .
Let us denote such an interval by I = [a , b ].
2 1 1
 a  b 1   a  b 1 
1
1
Now writing I = [a , b ] = a ,  , b 1  …(2)
2 1 1  1  
 2   2 
As argued before, at least one of the intervals in the union of (2) cannot be covered by a finite
sub-class of .
Let us denote such an interval by I = [a , b ].
3 2 2
On continuing this process we obtain a nested sequence I  of closed intervals such that none of
n
these intervals I can be covered by a finite sub-class of .
n
Clearly the length of the inverval.
a b

I =
n n
2
Thus lim |I | = 0.
n
Hence, by the nested closed interval property,  I  .
n
Let p   I , then p  I  n  N.
n n
In particular p  I .
1
Now since  is an open covering of I , there exists some A in C such that p A .
1  0  0
Since A is open there exists an open interval (p – , p + ) such that p  (p – , p + )  A .


0 0
Since (I )  0 as n  , there exists some
n
I  (p – , p + )  A .
n 
0 0
This contradicts our assumption that no I is covered by a finite number of members of .
n
Hence [a, b] is compact.

Example 4: The real line is not compact.
Solution: Let  = { ] –n, n [ : n  N}.

Then each member of  is clearly an open interval and therefore, a U-open set.
Also if p is any real number, then there exists a positive integer n such that n > |p|.
p p
Then clearly p  ] –n , n [  .
p p
Thus each point of  is contained in some member of  and therefore  is an open covering of .

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