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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 Suspension (topology) ： ウィキペディア英語版 Suspension (topology) In topology, the suspension ''SX'' of a topological space ''X'' is the quotient space: :$SX\; =\; (X\; \backslash times\; I)/\backslash \; x\_1,x\_2\; \backslash in\; X\backslash \}$ of the product of ''X'' with the unit interval ''I'' = (1 ). Thus, ''X'' is stretched into a cylinder and then both ends are collapsed to points. One views ''X'' as "suspended" between the end points. One can also view the suspension as two cones on ''X'' glued together at their base (or as a quotient of a single cone). Given a continuous map $f:X\backslash rightarrow\; Y,$ there is a map $Sf:SX\backslash rightarrow\; SY$ defined by $Sf(()):=().$ This makes $S$ into a functor from the category of topological spaces into itself. In rough terms ''S'' increases the dimension of a space by one: it takes an ''n''-sphere to an (''n'' + 1)-sphere for ''n'' ≥ 0. The space $SX$ is homeomorphic to the join $X\backslash star\; S^0,$ where $S^0$ is a discrete space with two points. The space $SX$ is sometimes called the unreduced, unbased, or free suspension of $X$, to distinguish it from the reduced suspension described below. The suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies. In homotopy theory, the phenomena which are preserved under suspension, in a suitable sense, make up stable homotopy theory. ==Reduced suspension== If ''X'' is a pointed space (with basepoint ''x''0), there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension Σ''X'' of ''X'' is the quotient space: :$\backslash Sigma\; X\; =\; (X\backslash times\; I)/(X\backslash times\backslash \backslash cup\; X\backslash times\backslash \backslash cup\; \backslash \backslash times\; I)$. This is the equivalent to taking ''SX'' and collapsing the line (''x''0 × ''I'') joining the two ends to a single point. The basepoint of Σ''X'' is the equivalence class of (''x''0, 0). One can show that the reduced suspension of ''X'' is homeomorphic to the smash product of ''X'' with the unit circle ''S''1. :$\backslash Sigma\; X\; \backslash cong\; S^1\; \backslash wedge\; X$ For well-behaved spaces, such as CW complexes, the reduced suspension of ''X'' is homotopy equivalent to the ordinary suspension. Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is a left adjoint to the functor $\backslash Omega$ taking a (based) space $X$ to its loop space $\backslash Omega\; X$. In other words, :$\backslash operatorname\_$ *\left(\Sigma X,Y\right)\cong \operatorname_ *\left(X,\Omega Y\right) naturally, where $\backslash operatorname\_$ *\left(X,Y\right) stands for continuous maps which preserve basepoints. This is not the case for unreduced suspension and free loop 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「Suspension (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.