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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 induced topology ： ウィキペディア英語版 induced topology In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space. == Definition == Let $X\_0,\; X\_1$ be sets, $f:X\_0\backslash to\; X\_1$. If $\backslash tau\_0$ is a topology on $X\_0$, then a topology coinduced on $X\_1$ by $f$ is $\backslash$. If $\backslash tau\_1$ is a topology on $X\_1$, then a topology induced on $X\_0$ by $f$ is $\backslash$. The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union and intersection. Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set $X\_0=\backslash$ with a topology $\backslash \backslash \}$, a set $X\_1=\backslash$ and a function $f:X\_0\backslash to\; X\_1$ such that $f(-2)=-1,\; f(-1)=0,\; f(1)=0,\; f(2)=1$. A set of subsets $\backslash tau\_1=\backslash$ is not a topology, because $\backslash \backslash \}\; \backslash subseteq\; \backslash tau\_1$ but $\backslash \; \backslash cap\; \backslash \; \backslash notin\; \backslash tau\_1$. There are equivalent definitions below. A topology $\backslash tau\_1$ induced on $X\_1$ by $f$ is the finest topology such that $f$ is continuous $(X\_0,\; \backslash tau\_0)\; \backslash to\; (X\_1,\; \backslash tau\_1)$. This is a particular case of the final topology on $X\_1$. A topology $\backslash tau\_0$ induced on $X\_0$ by $f$ is the coarsest topology such that $f$ is continuous $(X\_0,\; \backslash tau\_0)\; \backslash to\; (X\_1,\; \backslash tau\_1)$. This is a particular case of the initial topology on $X\_0$. 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「induced 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.