Page 178 - DMTH503_TOPOLOGY

Topology

Notes 18.6 Review Questions

1. Prove that regularity is a hereditary property.
2. Prove that normality is a topological property.

3. Prove that complete regularity is a topological property.
4. Show that if X is completely regular, then every pair of disjoint subsets A and B such that
A is compact and B is closed, there exists a real valued continuous mapping F of X such that
f(A) = {0} and f(B) = {1}.
5. Show that a closed subspace of a normal space is normal.
6. Show that a completely regular space is regular and hence a Tychonoff space is a T -space.
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7. Give an example of Hausdorff space which is not normal.
8. Show that a topological space X is normal iff for any closed set F and an open set G
containing F there exists an open set H such that

F H, H G i.e. F H  H G.

18.7 Further Readings

Books A.V. Arkhangel’skii, V.I. Ponomarev, Fundamentals of General Topology: Problems
and Exercises, Reidel (1984).

J.L. Kelly, General Topology, Springer (1975).
Stephen Willard (1970), General Topology.

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