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In mathematics, a Baire space is a topological space which has "enough" points that any 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. The space is named in honor of René-Louis Baire who introduced the concept.
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.
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.
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 closed 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.
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 in X if it is a union of countably many nowhere dense subsets
- of second category or nonmeagre 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 if its complement is meagre. A topological space X is a Baire space if and only if every comeager subset of X is dense.
Choice of Terminology
One can define denseness in the following sense:
A subset is dense at some point iff its closure is a neighborhood of that point.
One should compare this to the notion of closeness:
A subset is close to some point iff that point is contained in its closure.
So, similiraties become evident.
Now, this would give back precisely what is understood by a nowhere, somewhere and everywhere dense subset
However, there's a flaw in there since neither the complement of nowhere dense subset is everywhere dense in general nor the complement of an everywhere dense subset is nowhere dense in general.
So one should pay attention or rather use a terminology not related to denseness!
An example is given by. A subset is called small iff the interior of its closure is empty and large iff the closure of its interior is all. Note that the complement of a small subset is large and vice versa. However, being not small resp. not large does not imply being large resp. small in general. But that happens for various topological notions as for being close resp. open too.
Moreover, the same problem arises when considering the closure and dense subsets and their complement as well:
The complement of a dense subset is, well, not dense?!
This can be solved by rather considering the interior resp. exterior:
In this sense, the interior of the complement is empty whenever the exterior is empty and vice versa.
Dense subsets in this setting are precisely the ones with empty interior.
Note also, the only subset whose interior is all is the space itself and its complement the empty set is the only subset whose exterior is all.
Since being dense without the adverbials nowhere somewhere and everywhere is not contradicting itself, and since being dense is a commonly accepted notion, it would be beneficial to introduce a complementary notion especially in order to enlighten the interplay of Baire's Categories. This way in Baire Spaces, a countable intersection of large subsets (residual) is dense while analogously a countable union of small subsets (meagre) is complementary to being dense that is "sparse" in some sense.
While meagre seems intuitive its complementary notion is rather confusing: Residual!
A residual subset is one given by countable intersection of large subsets that is still big in some sense in Baire Spaces, namely dense. Therefore residual becomes counterintuitive. One should rather call it fat or at least adipose when following Baire's terminology.
- 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.
- 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.
- Here is an example of a set of second category in R with Lebesgue measure 0.
- Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.
Baire category theorem
- (BCT1) Every complete metric space is a Baire space. More generally, every topological space which 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 is a Baire space.
BCT1 shows that each of the following is a Baire space:
- The space R of real numbers
- The space of irrational numbers
- The Cantor set
- Indeed, every Polish space
- 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].
- Every open subspace of a Baire space is a Baire space.
- Given a family of continuous functions fn:X→Y with pointwise limit f:X→Y. If X is a Baire space then the points where f is not continuous is a meagre set 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.
- Munkres, James, Topology, 2nd edition, Prentice Hall, 2000.
- Baire, René-Louis (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1--123.