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In topology, two continuous functions from one topological space to another are called homotopic (Greek ὁμός (''homós'') = same, similar, and τόπος (''tópos'') = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. ==Formal definition== Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function from the product of the space ''X'' with the unit interval () to ''Y'' such that, if then and If we think of the second parameter of ''H'' as time then ''H'' describes a ''continuous deformation'' of ''f'' into ''g'': at time 0 we have the function ''f'' and at time 1 we have the function ''g''. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from ''f'' to ''g'' as the slider moves from 0 to 1, and vice versa. An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and and the map is continuous from to Y. The two versions coincide by setting It is not sufficient to require each map to be continuous.〔(Path homotopy and separately continuous functions )〕 The animation that is looped above right provides an example of a homotopy between two embeddings, ''f'' and ''g'', of the torus into . ''X'' is the torus, ''Y'' is , ''f'' is some continuous function from the torus to ''R''3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; ''g'' is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of ''h''t''(x)'' as a function of the parameter t, where t varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. 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「homotopy」の詳細全文を読む スポンサード リンク
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