11. The product topology used to define the Baire space can be described more concretely in terms of trees. 12. The representation of the Baire space as paths through a tree also gives a characterization of closed sets. 13. Conventionally, Baire space does not refer to this topology; it only refers to the product topology. 14. Therefore " R " is the complement of a subset of first category in a Baire space . 15. A "'Baire space "'is a topological space in which the interior has empty interior. 16. By definition of a Baire space , such sets ( called " residual sets " ) are dense. 17. The "'Wadge order "'is the preorder or quasiorder on the subsets of Baire space . 18. The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. 19. Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of integers. 20. The irrationals are a Baire space in the order topology, but rationals aren't, even though both are densely ordered.
