mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Kuratowski–Ulam theorem, introduced by , called also the Fubini theorem for

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

, is an analog of

Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the ...

for arbitrary

second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...

Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...

s. Let ''X'' and ''Y'' be second countable Baire spaces (or, in particular,

Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...

s), and let

A \subset X \times Y

. Then the following are equivalent if ''A'' has the

Baire property A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such t ...

: # ''A'' is meager (respectively comeager). # The

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

\

is comeager in X, where

A_x=\pi_Y \cap \lbrace x \rbrace \times Y /math>, where \pi_Y is the projection onto ''Y''.
Even if ''A'' does not have the Baire property, 2. follows from 1. Note that the theorem still holds (perhaps vacuously) for ''X'' an arbitrary

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

and ''Y'' a Hausdorff space with countable π-base. The theorem is analogous to the regular Fubini's theorem for the case where the considered

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

is a

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...

of a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

in a product space, with the usual correspondences, namely,

meagre set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...

with a set of measure zero, comeagre set with one of full measure, and a set with the Baire property with a measurable set.

References

* General topology Descriptive set theory Theorems in topology {{topology-stub