In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.
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In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets, smooth curves in the plane, and proper affine subspaces in a Euclidean space. A topological space is a Baire space if it is "large", meaning that it is not a countable union of negligible subsets. For example, the three dimensional Euclidean space is not a countable union of its affine planes.
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.
A topological space is called a Baire space if the union of any countable collection of closed sets with empty interior has empty interior.
This definition is equivalent to each of the following conditions:
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
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 if its complement is meagre. A topological space X is a Baire space if and only if every comeager subset of X is dense.
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in topology and functional analysis.
BCT1 shows that each of the following is a Baire space:
BCT2 shows that every manifold is a Baire space, even if it is not paracompact, and hence not metrizable. For example, the long line is of second category.