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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 Sorgenfrey plane ： ウィキペディア英語版 Sorgenfrey plane In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line $\backslash mathbb$ under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey. A basis for the Sorgenfrey plane, denoted $\backslash mathbb$ from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in $\backslash mathbb$ are unions of such rectangles. $\backslash mathbb$ is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal $\backslash Delta\; =\; \backslash$ is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space $\backslash mathbb$. It shows that separability does not inherit to closed subspaces. Note that $K\; =\; \backslash$ and $\backslash Delta\; \backslash setminus\; K$ are closed sets that cannot be separated by open sets, showing that $\backslash mathbb$ is not normal. Thus it serves as a counterexample to the notion that the product of normal spaces is normal; in fact, it shows that even the finite product of perfectly normal spaces need not be normal. ==References== * Reprinted as * Robert Sorgenfrey, "On the topological product of paracompact spaces", ''Bull. Amer. Math. Soc.'' 53 (1947) 631–632. * 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「Sorgenfrey plane」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.