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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 First-countable space ： ウィキペディア英語版 First-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space ''X'' is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point ''x'' in ''X'' there exists a sequence ''N''1, ''N''2, … of neighbourhoods of ''x'' such that for any neighbourhood ''N'' of ''x'' there exists an integer ''i'' with ''N''''i'' contained in ''N''. Since every neighborhood of any point contains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. ==Examples and counterexamples== The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''. An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line). Another counterexample is the ordinal space ω1+1 = () where ω1 is the first uncountable ordinal number. The element ω1 is a limit point of the subset [0,ω1) even though no sequence of elements in [0,ω1) has the element ω1 as its limit. In particular, the point ω1 in the space ω1+1 = [0,ω1] does not have a countable local base. Since ω1 is the only such point, however, the subspace ω1 = [0,ω1) is first-countable. The quotient space $\backslash mathbb/\backslash mathbb$ where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a Fréchet-Urysohn space. First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable. 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「First-countable space」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.