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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 Pasting lemma ： ウィキペディア英語版 Pasting lemma In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. For example in the book ''Topology and Groupoids'', where the condition given for the statement below is that $A\; \backslash setminus\; B\; \backslash subseteq\; \backslash operatorname\; A$ and $B\; \backslash setminus\; A\; \backslash subseteq\; \backslash operatorname\; B$. The pasting (or gluing) lemma is crucial to the construction of the fundamental group or fundamental groupoid of a topological space; it allows one to concatenate continuous paths to create a new continuous path. == Formal statement == Let $X,Y$ be both closed (or both open) subsets of a topological space ''A'' such that $A\; =\; X\; \backslash cup\; Y$, and let ''B'' also be a topological space. If $f:\; A\; \backslash to\; B$ is continuous when restricted to both ''X'' and ''Y'', then ''f'' is continuous. This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one. Proof: if ''U'' is a closed subset of ''B'', then $f^(U\; )\backslash cap\; X$ and $f^(U\; )\backslash cap\; Y$ are both closed since each is the preimage of ''f'' when restricted to ''X'' and ''Y'' respectively, which by assumption are continuous. Then their union, $f^(U)$ is also closed, being a finite union of closed sets. A similar argument applies when ''X'' and ''Y'' are both open. $\backslash Box$ The infinite analog of this result (where $A=X\_1\backslash cup\; X\_2\backslash cup\; X\_3\backslash cup\backslash cdots$)is not true for closed $X\_1,\; X\_2,\; X\_3\backslash ldots$. (For instance, the inclusion map $\backslash iota:Z\backslash rightarrow\; R$ from the integers to the real line (with the integers equipped with the cofinite topology) is continuous when restricted to an integer, but the inverse image of a bounded open set in the reals with this map is at most a finite number of points, so not open in ''Z''.) It is, however, true if the $X\_1,\; X\_2,\; X\_3\backslash ldots$ are open; this follows from the fact that an arbitrary union of open sets is open. 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「Pasting lemma」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.