The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. A topological space is called a Baire space if it satisfies any of the following equivalent conditions:
Another large class of Baire spaces are algebraic varieties with the Zariski topology. For example, the space of complex numbers whose open sets are complements of the vanishing sets of polynomials is an algebraic variety with the Zariski topology. Usually this is denoted .
The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval with the usual topology.
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.
One of the first non-examples comes from the induced topology of the rationals inside of the real line with the standard euclidean topology.
Given an indexing of the rationals by the natural numbers so a bijection and let where which is an open, dense subset in
Then, because the intersection of every open set in is empty, the space cannot be a Baire space.
Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of 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
Köthe, Gottfried (1983) . Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.