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Unit 31: Baire Spaces

exist no subsets of Z having empty interior, except for the empty set. Therefore Z satisfies the Notes
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Baire condition vacuously.
Lemma 1: X is a Baire space iff gives any countable collection { } of open sets in X, each of which
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is dense in X their intersection is also dense in X.
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Proof: Recall that a set  is dense in X if C = X. The theorem now follows at once from the two
remarks.
1. A is closed in X iff X-A is open in X.
2. B has empty interior in X if and only if X-B is dense in X.
Lemma 2: Any open subspace Y of a Baire space X is itself a Baire space.

Proof: Let A be a countable collection of closed set of Y that have empty interiors in Y. We show
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that A has empty interior in Y.
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Let A be the closure of A in X; then A  Y = A . The set A has empty interior in X. For it
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is a non empty open set of X contained in A , then must intersect A . Then  Y is a
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non-empty open set of Y contained in A , contrary to hypothesis.
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If the union of the sets A contains the non empty open set W of Y, then the union of the sets
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A also contains the set W, which is open in X because Y is open in X. But each set A has empty
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interior in X, contradicting the fact that X is a Baire space.
31.1.2 Baire’s Category Theory
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.
Definition: A metric space is said to be totally of second category if every non empty closed
subset of X is of the second category.

Example 2: Let q be arbitrary.

{q} = {q}  D ({q}), [ A = A  D(A) ]
= {q}   = {q}
 int {q} = int {q}
=  {G   : G is open, G  {q}} = .
For every subset of  contains rational as well irrational numbers.
Thus, int {q} = .

This proves that {q} is a non-dense subset of .
 =  {{q} : a  }.
Furthermore  is enumerable.
  is an enumerable union of non-dense sets.
From what has been done it follows that  is of the first category.

Example 3: Consider a sequence <f (x)> of continuous functions defined from I = [0, 1]
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into  s.t. f (x) = x  x  N.
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