翻訳と辞書 Words near each other ・ Isolate Records ・ Isolated 17,20-lyase deficiency ・ Isolated Atoms ・ Isolated atrial amyloidosis ・ Isolated brain ・ Isolated congenital asplenia ・ Isolated danger mark ・ Isolated ground ・ Isolated hepatic perfusion ・ Isolated horizon ・ Isolated hypogonadotropic hypogonadism ・ Isolated Incident ・ Isolated lung perfusion ・ Isolated Mass/Circumflex (Number 2) ・ Isolated pawn ・ Isolated point ・ Isolated Power ・ Isolated primary immunoglobulin M deficiency ・ Isolated singularity ・ Isolated system ・ Isolated Warrior ・ Isolated-phase bus ・ Isolates ・ Isolating language ・ Isolating neighborhood ・ Isolation ・ Isolation (2005 film) ・ Isolation (2009 film) ・ Isolation (Alter Bridge song) ・ Isolation (Carpathian album) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Isolated point ： ウィキペディア英語版 Isolated point ] In topology, a branch of mathematics concerning the study of shapes and spaces, a point ''x'' of a topological space ''X'' is called an isolated point of a subset ''S'' of ''X'' if ''x'' belongs to ''S'' and there exists in ''X'' a neighborhood of ''x'' not containing other points of ''S''. This is equivalent to saying that the singleton is an open set in the topological space ''S'' (considered as a subspace of ''X''). In particular, in a Euclidean space (or in any other metric space), ''x'' is an isolated point of ''S'', if one can find an open ball around ''x'' which contains no other points of ''S''. Equivalently, a point ''x'' in ''S'' is an isolated point of ''S'' if and only if it is not a limit point of ''S''. A set which is made up only of isolated points is called a discrete set. Any discrete subset of Euclidean space is countable, since the isolation of each of its points (together with the fact the rationals are dense in the reals) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many. However, a set can be countable but not discrete, e.g. the rational numbers with the absolute difference metric. See also discrete space. A set with no isolated point is said to be dense-in-itself. A closed set with no isolated point is called a perfect set. The number of isolated points is a topological invariant, i.e. if two topological spaces $X$ and $Y$ are homeomorphic, the number of isolated points in each is equal. ==Examples== Topological spaces in the following examples are considered as subspaces of the real line with the standard topology. * For the set $S=\backslash \backslash cup\; (2)$, the point 0 is an isolated point. * For the set $S=\backslash \backslash cup\; \backslash$, each of the points 1/k is an isolated point, but 0 is not an isolated point because there are other points in ''S'' as close to 0 as desired. * The set $=\; \backslash$ of natural numbers is a discrete set. * The Morse lemma states that non-degenerate critical points of certain functions are isolated. 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「Isolated point」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.