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In mathematics, a path in a topological space ''X'' is a continuous function ''f'' from the unit interval ''I'' = () to ''X'' :''f'' : ''I'' → ''X''. The ''initial point'' of the path is ''f''(0) and the ''terminal point'' is ''f''(1). One often speaks of a "path from ''x'' to ''y''" where ''x'' and ''y'' are the initial and terminal points of the path. Note that a path is not just a subset of ''X'' which "looks like" a curve, it also includes a parameterization. For example, the maps ''f''(''x'') = ''x'' and ''g''(''x'') = ''x''2 represent two different paths from 0 to 1 on the real line. A loop in a space ''X'' based at ''x'' ∈ ''X'' is a path from ''x'' to ''x''. A loop may be equally well regarded as a map ''f'' : ''I'' → ''X'' with ''f''(0) = ''f''(1) or as a continuous map from the unit circle ''S''1 to ''X'' :''f'' : ''S''1 → ''X''. This is because ''S''1 may be regarded as a quotient of ''I'' under the identification 0 ∼ 1. The set of all loops in ''X'' forms a space called the loop space of ''X''. A topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components. The set of path-connected components of a space ''X'' is often denoted π0(''X'');. One can also define paths and loops in pointed spaces, which are important in homotopy theory. If ''X'' is a topological space with basepoint ''x''0, then a path in ''X'' is one whose initial point is ''x''0. Likewise, a loop in ''X'' is one that is based at ''x''0. ==Homotopy of paths== (詳細はalgebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed. Specifically, a homotopy of paths, or path-homotopy, in ''X'' is a family of paths ''f''''t'' : ''I'' → ''X'' indexed by ''I'' such that * ''f''''t''(0) = ''x''0 and ''f''''t''(1) = ''x''1 are fixed. * the map ''F'' : ''I'' × ''I'' → ''X'' given by ''F''(''s'', ''t'') = ''f''''t''(''s'') is continuous. The paths ''f''0 and ''f''1 connected by a homotopy are said to homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed. The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path ''f'' under this relation is called the homotopy class of ''f'', often denoted (). 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「Path (topology)」の詳細全文を読む スポンサード リンク
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