Page 169 - DMTH503_TOPOLOGY

Unit 17: The Separation Axioms

5. For any set X, there exists a unique smallest topology T such that (X, T) is a T -space. Notes
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6. A T -space is countably compact iff every infinite open covering has a proper subcover.
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7. If (X, T) is a T -space and T*  T, then (X, T*) is also a T -space.
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8. If (X, T ) is a Hausdorff space, (X, T ) is compact and T  T than T = T .
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9. If f and g are continuous mappings of a topological space X into a Hausdorff space, then the
set of points at which f and g are equal is a closed subset of X.
10. If f is a continuous mapping of a Hausdorff space X into itself, show that the set of fixed
points; i.e. {x : f(x) = x}, is closed.
11. Show that every infinite Hausdorff space contains an infinite isolated set.
12. If (X, T) is a T -space and T*  T, then prove that (X, T*) is also a T -space.
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17.6 Further Readings

Books Eric Schechter (1997), Handbook of Analysis and its Foundations, Academic Press.
Stephen Willard, General Topology, Addison Wesley, 1970 reprinted by Dover
Publications, New York, 2004.

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