User:PatrickR2/sandbox2

In mathematics, a topological space $$X$$ is said to be a Baire space, if for any given countable collection $$\{A_n\}$$ of closed sets with empty interior in $$X$$, their union $\bigcup_n A_n$ also has empty interior in $$X$$. Equivalently, a locally convex space which is not meagre in itself is called a Baire space. According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space. Bourbaki coined the term "Baire space".

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 $$\R,$$ 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. A topological space $$X$$ is called a Baire space if it satisfies any of the following equivalent conditions:  Every intersection of countably many dense open sets in $$X$$ is dense in $$X$$; Every union of countably many closed subsets of $$X$$ with empty interior has empty interior; Every non-empty open subset of $$X$$ is a nonmeager subset of $$X$$; Every comeagre subset of $$X$$ is dense in $$X$$; Whenever the union of countably many closed subsets of $$X$$ has an interior point, then at least one of the closed subsets must have an interior point; Every point in $$X$$ has a neighborhood that is a Baire space (according to any defining condition other than this one). 
 * So $$X$$ is a Baire space if and only if it is "locally a Baire space."

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) Every complete pseudometric space is a Baire space. More generally, every topological space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space. In particular, every completely metrizable space is a Baire space.
 * (BCT2) Every locally compact Hausdorff space (or more generally every locally compact sober space) is a Baire space.

BCT1 shows that each of the following is a Baire space:

 The space $$\R$$ of real numbers</li> <li>The space of irrational numbers, which is homeomorphic to the Baire space $\omega^{\omega}$ of set theory</li> <li>Every compact Hausdorff space is a Baire space. <li>Indeed, every Polish space.</li> </ul>
 * In particular, the Cantor set is a Baire space.</li>

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.

Other sufficient conditions
<ul> <li>A product of complete metric spaces is a Baire space.</li> <li>A topological vector space is nonmeagre if and only if it is a Baire space, which happens if and only if every closed balanced absorbing subset has non-empty interior.</li> </ul>

Examples
<ul> <li>The space $$\R$$ of real numbers with the usual topology, is a Baire space, and so is of second category in itself. The rational numbers are of first category and the irrational numbers are of second category in $$\R$$.</li> <li>Another large class of Baire spaces are algebraic varieties with the Zariski topology. For example, the space $$\mathbb{C}^n$$ of complex numbers whose open sets are complements of the vanishing sets of polynomials $$f \in \mathbb{C}[x_1,\ldots,x_n]$$ is an algebraic variety with the Zariski topology. Usually this is denoted $$\mathbb{A}^n$$. </li> <li>The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval $$[0, 1]$$ with the usual topology.</li> <li>Here is an example of a set of second category in $$\R$$ with Lebesgue measure $$0$$:
 * $$\bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty} \left(r_{n}-\left(\tfrac{1}{2}\right)^{n+m}, r_{n}+\left(\tfrac{1}{2}\right)^{n+m}\right)$$

where $$\left(r_n\right)_{n=1}^{\infty}$$ is a sequence that enumerates the rational numbers.</li> <li>Note that the space of rational numbers with the usual topology inherited from the real numbers is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.</li> </ul>

Non-example
One of the first non-examples comes from the induced topology of the rationals $$\Q$$ inside of the real line $$\R$$ with the standard euclidean topology. Given an indexing of the rationals by the natural numbers $$\N$$ so a bijection $$f : \N \to \Q,$$ and let $$\mathcal{A} = \left(A_n\right)_{n=1}^{\infty}$$ where $$A_n := \Q \setminus \{ f(n) \},$$ which is an open, dense subset in $$\Q.$$ Then, because the intersection of every open set in $$\mathcal{A}$$ is empty, the space $$\Q$$ cannot be a Baire space.

Properties
<ul> <li>Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of $$X$$ is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval $$[0, 1].$$</li> <li>Every open subspace of a Baire space is a Baire space.</li> <li>Given a family of continuous functions $$f_n : X \to Y$$= with pointwise limit $$f : X \to Y.$$ If $$X$$ is a Baire space then the points where $$f$$ is not continuous is in $$X$$ and the set of points where $$f$$ is continuous is dense in $$X.$$ A special case of this is the uniform boundedness principle.</li> <li>A closed subset of a Baire space is not necessarily Baire.</li> <li>The product of two Baire spaces is not necessarily Baire. However, there exist sufficient conditions that will guarantee that a product of arbitrarily many Baire spaces is again Baire.</li> </ul>