Page 230 - DMTH503_TOPOLOGY

Topology

Notes X is a metric spaceX is a Hausdroff space w.r.t. the metric topology.
Being a compact subset of a Hausdroff space, A is closed.
A is compact. A is sequentially compact.
 A is totally bounded.
Finally, we have shown that A is closed and totally bounded.

Conversely, suppose that A is closed and totally bounded subset of complete metric space (X, d).
To prove that A is compact.
A is complete, being a closed subset of a complete metric space (X, d). Thus A is complete and
totally bounded.

Self Assessment

3. Let X be a metric space and Y is a complete metric space, and let A be dense subspace of X.
If f is a uniformly continuous mapping of A into Y, then f can be extended uniquely to a
uniformly continuous map of X into Y.
4. Let A be subspace of a complete metric space and show that A is compact  A is totally
founded.
5. If <A > is a sequence of nowhere dense sets in a complete metric space X, then there exist
n
a point in X which is not in any of the A ’s.
n
27.4 Summary

 A sequence <x > is Cauchy if given> 0,a positive integer n such that
n o
d(x , x ) <  for all n  n and for all p1.
n+p n o
 A metric space X is said to be complete if every Cauchy sequence of points in X converges
to a point in X.
 A metric space is compact iff it is totally bounded and complete.

27.5 Keywords

Closed Set: A set A is said to be closed if every limiting point of A belongs to the set A itself.

Cluster Point: Let (X, T) be a topological space and AX. A point xX is said to be the cluster
point if each open set containing x contains at least one point of A different from x.

Convergent Sequence: A sequence <a > is said to converge to a, if  > 0,n N, s.t. nn 
n o o
-
a a < .
n
Sequentially Compact: A metric space (X, d) is said to be sequentially compact if every sequence
in X has a convergent subsequence.

27.6 Review Questions

1. If a Cauchy sequence has a convergent subsequence, then prove that it is itself convergent.
2. Show that every compact metric space is complete.

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