Baire space

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. 
Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. 
The spaces are named in honor of René-Louis Baire who introduced the concept.
Motivation.
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 in formula_1 smooth curves in the plane, and proper affine subspaces in a Euclidean space. 
If a topological space is a Baire space then 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.
Definition.
The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. This article first gives some preliminary definitions that Baire used in his original work, and afterwards introduces some modern equivalent definitions. 
Definitions.
In his original definition, Baire defined a notion of category (unrelated to category theory) as follows.
A subset formula_2 of a topological space formula_3 is called nowhere dense or rare if its closure in formula_3 has empty interior in formula_3; that is, if formula_6 Importantly, a closed subset of formula_3 is nowhere dense if and only if its interior in formula_3 is empty. The closures and interiors in this definition are always taken relative to formula_3 rather than formula_2 because to do otherwise would result in a useless definition (specifically, this is because formula_11 and formula_12 are always true for every formula_13 only formula_14 satisfies formula_15). 
A subset of a topological space formula_3 is said to be meagre in formula_17 a meagre sub of formula_17 or of the first category in formula_3 if it is equal to a countable union of nowhere dense subsets of formula_20 A subset is of the second category or nonmeagre in formula_3 if it is not of first category in formula_20 
A topological space is called meagre (resp. nonmeagre) if it is a meagre (resp. nonmeagre) subset of itself.
A subset formula_33 of formula_3 is comeagre in formula_3 if its complement formula_36 is meagre in formula_20 
Baire space definition.
A topological space formula_3 is called a Baire space if it satisfies any of the following equivalent conditions:
Sufficient conditions.
Baire category theorem.
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.
Examples.
Non-example.
One of the first non-examples comes from the induced topology of the rationals formula_39 inside of the real line formula_40 with the standard euclidean topology. 
Given an indexing of the rationals by the natural numbers formula_41 so a bijection formula_42 and let formula_43 where formula_44 which is an open, dense subset in formula_45 
Then, because the intersection of every open set in formula_46 is empty, the space formula_39 cannot be a Baire space.