Baire space: Encyclopedia II - Baire space - Definition

Baire space - Definition

The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition which is closer to the definition originally given by Baire.

Baire space - Modern definition

A topological space is called a Baire space if the countable union of any collection of closed sets with empty interior has empty interior.

This definition is equivalent to each of the following conditions:

• Every intersection of countably many dense open sets is dense.
• The interior of every union of countably many nowhere dense sets is empty.
• Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.

Baire space - Historical definition

In his original definition, Baire defined a notion of category (unrelated to category theory) as follows

A subset of a topological space X is called

• nowhere dense in X if the interior of its closure is empty
• of first category or meagre (meager) in X if it is a union of countably many nowhere dense subsets
• of second category or nonmeagre (nonmeager) in X if it is not of first category in X

The definition for a Baire space can then be stated as follows: a topological space X is a Baire space if every non-empty open set is of second category in X. This definition is equivalent to the modern definition.

A subset A of X is comeagre (comeager) if its complement is meagre.

Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki