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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 Topological indistinguishability ： ウィキペディア英語版 Topological indistinguishability In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''A'' is the set of all neighborhoods that contain ''x'', and ''B'' is the set of all neighborhoods that contain ''y'', then ''x'' and ''y'' are "topologically indistinguishable" if and only if ''A'' = ''B''. Intuitively, two points are topologically indistinguishable if the topology of ''X'' is unable to discern between the points. Two points of ''X'' are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms. Topological indistinguishability defines an equivalence relation on any topological space ''X''. If ''x'' and ''y'' are points of ''X'' we write ''x'' ≡ ''y'' for "''x'' and ''y'' are topologically indistinguishable". The equivalence class of ''x'' will be denoted by (). ==Examples== For T0 spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one must look to non-T0 spaces to find interesting examples. On the other hand, regularity and normality do not imply T0, so we can find examples with these properties. In fact, almost all of the examples given below are completely regular. *In an indiscrete space, any two points are topologically indistinguishable. *In a pseudometric space, two points are topologically indistinguishable if and only if the distance between them is zero. *In a seminormed vector space, ''x'' ≡ ''y'' if and only if ‖''x'' − ''y''‖ = 0. * *For example, let ''L''2(R) be the space of all measurable functions from R to R which are square integrable (see ''L''''p'' space). Then two functions ''f'' and ''g'' in ''L''2(R) are topologically indistinguishable if and only if they are equal almost everywhere. *In a topological group, ''x'' ≡ ''y'' if and only if ''x''−1''y'' ∈ cl where cl is the closure of the trivial subgroup. The equivalence classes are just the cosets of cl (which is always a normal subgroup). *Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, ''x'' ≡ ''y'' if and only if the pair (''x'', ''y'') belongs to every entourage. The intersection of all the entourages is an equivalence relation on ''X'' which is just that of topological indistinguishability. *Let ''X'' have the initial topology with respect to a family of functions $\backslash$. Then two points ''x'' and ''y'' in ''X'' will be topologically indistinguishable if the family $f\_\backslash alpha$ does not separate them (i.e. $f\_\backslash alpha(x)\; =\; f\_\backslash alpha(y)$ for all $\backslash alpha$). *Given any equivalence relation on a set ''X'' there is a topology on ''X'' for which the notion of topological indistinguishability agrees with the given equivalence relation. One can simply take the equivalence classes as a base for the topology. This is called the partition topology on ''X''. 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「Topological indistinguishability」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.