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In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone (1936), and thus named in his honor. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. ==Stone spaces== Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a (closed) basis consisting of all sets of the form : where ''b'' is an element of ''B''. For every Boolean algebra ''B'', ''S''(''B'') is a compact totally disconnected Hausdorff space; such spaces are called Stone spaces (also ''profinite spaces''). Conversely, given any topological space ''X'', the collection of subsets of ''X'' that are clopen (both closed and open) is a Boolean algebra. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stone's representation theorem for Boolean algebras」の詳細全文を読む スポンサード リンク
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