Baire category theorem BaireCategoryTheorem 2002-06-04 07:41:44 2004-09-29 11:27:00 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} In a non-empty complete metric space, any countable intersection of dense, open subsets is non-empty. In fact, such countable intersections of dense, open subsets are dense. So the theorem holds also for any non-empty open subset of a complete metric space. \textbf{Alternative formulations:} Call a set \emph{first category}, or a \emph{meagre} set, if it is a countable union of nowhere dense sets, otherwise \emph{second category}. The Baire category theorem is often stated as ``no non-empty complete metric space is of first category'', or, trivially, as ``a non-empty, complete metric space is of second category''. In short, this theorem says that every nonempty complete metric space is a Baire space. In functional analysis, this important property of complete metric spaces forms the {b}asis for the proofs of the important principles of Banach spaces: the open mapping theorem and the closed graph theorem. It may also be taken as giving a concept of ``small sets'', similar to sets of measure zero: a countable union of these sets remains ``small''. However, the real line $\mathbb{R}$ may be partitioned into a set of measure zero and a set of first category; the two concepts are distinct. Note that, apart from the requirement that the set be a complete metric space, all conditions and conclusions of the theorem are phrased topologically. This ``metric requirement'' is thus something of a disappointment. As it turns out, there are two ways to reduce this requirement. First, if a topological space $\mathcal{T}$ is homeomorphic to a non-empty open subset of a complete metric space, then we can transfer the Baire property through the homeomorphism, so in $\mathcal{T}$ too any countable intersection of open dense sets is non-empty (and, in fact, dense). The other formulations also hold in this case. Second, the Baire category theorem holds for a locally compact, Hausdorff\footnote{Some authors only define a locally compact space to be a Hausdorff space; that is the sense required for this theorem.} topological space $\mathcal{T}$.