Page 143 - DMTH503_TOPOLOGY

Unit 14: Limit Point Compactness

14.4 Review Questions Notes

1. Show that [0, 1] is not limit point compact as a subspace of .
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2. Let X be limit point compact.
(a) If f : X Y is continuous, does it follow that f(X) is limit point compact?
(b) If A is a closed subset of X, does it follow that A is limit point compact?
(c) If X is a subspace of the Hausdorff space Z, does it follow that X is closed in Z?
3. A space X is said to be countably compact if every countable open covering of X contains
a finite subcollection that covers X. Show that for a T space X, countable compactness is
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equivalent to limit point compactness.
[Hint: If no finite subcollection of U covers X, choose x   U 1 ... U , for each n.]
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14.5 Further Readings

Books J.L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York.
S. Willard, General Topology, Addison-Wesley Mass. 1970.

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