Page 137 - DMTH503_TOPOLOGY

Unit 13: Compact Spaces and Compact Subspace of Real Line

Self Assessment Notes

4. Prove that if X is an ordered set in which every closed interval is compact, the X has the
least upper bound property.

5. Let X be a metric space with metric d; let A X be non-empty. Show that d(x, A) = 0 if and
only if x A.

13.3 Summary

 A collection A of subsets of a space X is said to cover X, or to be a covering of X, if the union
of the elements of A is equal to X. It is called an open covering of X if its element are ope
subsets of X.

 A space X is said to be compact if every open covering A of X contains a finite subcollection
that also cover X.

 Let A be an open covering of the metric space (X, d). If X is compact, there is a > 0 such that
for each subset of X having diameter less than , there exists an element of A continuing it.
The number  is called a Lebesgue number for the covering A.

13.4 Keywords

Closed Open Set: Let (X, T) be a topological space. Any set A T is called an open set and X – A
is a closed set.
Countably Compact: A topological space (X, T) is said to be countably compact iff every countable
T-open cover of X has a finite subcover.
Homeomorphism: A map f : (X, T) (Y, U) is said to be homeomorphism if (i) f is one-one onto
–1
(ii) f and f are continuous.
Indiscrete Topology: Let X be any non-empty set ad T = {X, }. Then T is called the indiscrete
topology.

13.5 Review Questions

1. Let T and T be two topologies on the set X; suppose that TT. What does compactness of
X under one of these topologies imply about compactness under the other?
2. Show that if X is compact Hausdorff under both T and T, then either T and T are equal or
they are not comparable.
3. Show that a finite union of compact subspaces of X is compact.

4. Let A and B be disjoint compact subspace of the Hausdorff space X. Show that there exist
disjoint open sets U and V containing A and B, respectively.

5. Let Y be a subspace of X. If Z Y, then show that Z is compact as a subspace of Y it is
compact as a subspace of X.
6. Prove that a closed subset of a compact space is compact.

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