Page 229 - DMTH503_TOPOLOGY

Unit 27: Complete Metric Spaces

Theorem 4: A metric space is compact iff it is totally bounded and complete. Notes
Proof: Let (X, d) be a compact metric space.
To prove that X is complete and totally bounded.
X is compact. X is sequentially compact. …(1)

 X is totally bounded. …(2)
X is sequentially compact.  every sequence in X has convergent subsequence.
 In particular, every Cauchy sequence in X has a convergent subsequence
 Every Cauchy sequence in X converges to some point in X.

 X is complete. …(3)
From (2) & (3) the required result follows.
Conversely, suppose that a metric space (X, d) is complete and totally bounded.
To prove that X is compact.

Consider an arbitrary sequence
S = <x , x , x , …>
1 11 12 13
X is totally boundedfinite class of open spheres, each of radius 1, whose union is X.
From this we can deduce that S has a subsequence
1
S = <x , x , x , …>
2 21 22 23
1
all of whose points be in some open sphere of radius .
2
Similarly we can construct a subsequence S of S s.t.
3 2
S = <x , x , x , …>
3 31 32 33
1
all of whose points be in some open sphere of radius .
3
We continue this process to from successive subsequences. Now we suppose that

S = <x , x , x , …>.
11 22 33
Then S is a diagonal subsequence to form successive subsequence. Now we suppose that
S = <x , x , x ...>. Then S is a diagonal subsequence of S . By nature of this construction, S is
11 22 33 1
clearly Cauchy subsequence of S .
1
X is completeevery Cauchy sequence in X is convergent.
in particular, the Cauchy sequence S is convergent.

Finally, the sequence, S has a convergent subsequence S. Since the sequence S in X is arbitrary
1 1
and hence every sequence in X has a convergent subsequence, meaning thereby X is sequentially
compact and hence X is compact.
Theorem 5: Let A be a subset of a complete metric space (X, d). Prove that A is compactA is
closed and totally bounded.
Proof: Let A be a compact subset of complete metric space (X, d).

To prove that A is closed and totally bounded.

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