|
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 because they were first extensively studied by Polish topologists and logicians — Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory. Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish. Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the cardinality of the continuum. Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces. ==Properties== # (Alexandrov's theorem) If ''X'' is Polish then so is any ''G''δ subset of ''X''. # (Cantor–Bendixson theorem) If ''X'' is Polish then any closed subset of ''X'' can be written as the disjoint union of a perfect subset and a countable open subset. # A subspace ''Q'' of a Polish space ''P'' is Polish if and only if ''Q'' is the intersection of a sequence of open subsets of ''P''. (This is the converse to Alexandrov's theorem.) # A topological space ''X'' is Polish if and only if ''X'' is homeomorphic to the intersection of a sequence of open subsets of the cube , where ''I'' is the unit interval and ''N'' is the set of natural numbers. The following spaces are Polish: * closed subsets of a Polish space, * open subsets of a Polish spaces * products and disjoint unions of countable families of Polish spaces, * locally compact spaces that are metrizable and countable at infinity, * countable intersections of Polish subspaces of a Hausdorff topological space, * the set of nonrational numbers with the topology induced by the real line. 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「Polish space」の詳細全文を読む スポンサード リンク
|