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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 Lower limit topology ： ウィキペディア英語版 Lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R (generated by the open intervals) and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [''a'',''b''), where ''a'' and ''b'' are real numbers. The resulting topological space, sometimes written R''l'' and called the Sorgenfrey line after Robert Sorgenfrey, often serves as a useful counterexample in general topology, like the Cantor set and the long line. The product of R''l'' with itself is also a useful counterexample, known as the Sorgenfrey plane. In complete analogy, one can also define the upper limit topology, or left half-open interval topology. == Properties == * The lower limit topology is finer (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a countably infinite union of half-open intervals. * For any real $a$ and $b$, the interval $\backslash cap\backslash mathbb$. Since the intervals $(a(x),\; x]$, parametrized by $x\; \backslash in\; C$, are pairwise disjoint, the function $q:\; C\; \backslash to\; \backslash mathbb$ is injective, and so $C$ is at most countable. * The name "lower limit topology" comes from the following fact: a sequence (or net) $(x\_\backslash alpha)$ in $\backslash mathbb\_l$ converges to the limit $L$ iff it "approaches $L$ from the right", meaning for every $\backslash epsilon>0$ there exists an index $\backslash alpha\_0$ such that . The Sorgenfrey line can thus be used to study right-sided limits: if $f:\; \backslash mathbb\; \backslash to\; R$ is a function, then the ordinary right-sided limit of $f$ at $x$ (when the codomain carry the standard topology) is the same as the usual limit of $f$ at $x$ when the domain is equipped with the lower limit topology and the codomain carries the standard topology. * In terms of separation axioms, $\backslash mathbb\_l$ is a perfectly normal Hausdorff space. * In terms of countability axioms, $\backslash mathbb\_l$ is first-countable and separable, but not second-countable. * In terms of compactness properties, $\backslash mathbb\_l$ is Lindelöf and paracompact, but not σ-compact nor locally compact. * $\backslash mathbb\_l$ is not metrizable, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a premetric. * $\backslash mathbb\_l$ is a Baire space (). 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「Lower limit topology」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.