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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pointclass ： ウィキペディア英語版 Pointclass In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of ''definability property''; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.) Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability), the property of Baire, and the perfect set property. ==Basic framework== In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire space or sometimes Cantor space, each of which has the advantage of being zero dimensional, and indeed homeomorphic to its finite or countable powers, so that considerations of dimensionality never arise. Moschovakis provides greater generality by fixing once and for all a collection of underlying Polish spaces, including the set of all naturals, the set of all reals, Baire space, and Cantor space, and otherwise allowing the reader to throw in any desired perfect Polish space. Then he defines a ''product space'' to be any finite Cartesian product of these underlying spaces. Then, for example, the pointclass $\backslash boldsymbol^0\_1$ of all open sets means the collection of all open subsets of one of these product spaces. This approach prevents $\backslash boldsymbol^0\_1$ from being a proper class, while avoiding excessive specificity as to the particular Polish spaces being considered (given that the focus is on the fact that $\backslash boldsymbol^0\_1$ is the collection of open sets, not on the spaces themselves). 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「Pointclass」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.