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Topology

Notes 31.2 Summary

 A space X is said to be a Baire space if the following condition holds: Given any countable
collection {A } of closed sets of X each of which has empty interior in X, their unionA
n n
also has empty interior in X.
 Let (X, d) be a metric space and AX. The set A is called of the first category if it can be
expressed as a countable union of non dense sets. The set A is called of the second category
if it is not of the first category.

31.3 Keywords

Complete Metric Space: A metric space X is said to be complete if every Cauchy sequence of
points in X converges to a point in X.
Dense: A said to be dense in X if A = X.

Nowhere Dense: A is said to be nowhere dense if ( A )° = .
31.4 Review Questions

1. Show that if every point x of X has a neighborhood that is a Baire space, then X is a Baire
space.
[Hint: Use the open set formulation of the Baire Condition].

2. Show that every locally compact Hausdorff space is a Baire space.
3. Show that the irrationals are a Baire space.
4. A point x in a topological space (X, T) is said to be an isolated point if {x}  T. Prove if (X, T)
is a countable T -space with no isolated points. Then it is not a Baire space.
1
5. Let (X, T) be any topological space and Y and S dense subsets of X. If S is also open in (X, T),
prove that SY is dense in both X and Y.
6. Let (X, T) and (Y, T ) be topological space and f : (X, T)(Y, T ) be a continuous open
1 1
mapping. If (X, T) is a Baire space. Show that an open continuous image of a Baire space is
a Baire space.
7. Let (Y, T ) be an open subspace of the Baire space (X, T). Prove that (Y, T) is a Baire space.
1
So an open subspace of a Baire space is a Baire space.
8. Let B be a Banach space where the dimension of the underlying vector space is countable.
Using the Baire Category Theorem, prove that the dimension of the underlying vector
space is, in fact, finite.

31.5 Further Readings

Books A.V. Arkhangal’skii, V.I. Ponomarev, Fundamentals of General Topology: Problems
and Exercises, Reidel (1984).
J. Dugundji, Topology, Prentice Hall of India, New Delhi.

Online link www.springer.com/978-3642-00233-5

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