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单词 Path
释义

path


Let I=[0,1] and let X be a topological spaceMathworldPlanetmath.

A continuous mapMathworldPlanetmath f:IX such that f(0)=x and f(1)=y is called a path in X. The point x is called the initial point of the path and y is called its terminal point. If, in addition, the map is one-to-one, then it is known as an arc.

Sometimes, it is convenient to regard two paths or arcs as equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if they differ by a reparameterization. That is to say, we regard f:IX and g:IX as equivalent if there exists a homeomorphism h:II such that h(0)=0 and h(1)=1 and f=gh.

If the space X has extra structureMathworldPlanetmath, one may choose to restrict the classes of paths and reparameterizations. For example, if X has a differentiable structure, one may consider the class of differentiableMathworldPlanetmathPlanetmath paths. Likewise, one can speak of piecewise linear paths, rectifiable paths, and analyticPlanetmathPlanetmath paths in suitable contexts.

The space X is said to be pathwise connected if, for every two points x,yX, there exists a path having x as initial point and y as terminal point. Likewise, the space X is said to be arcwise connected if, for every two distinct points x,yX, there exists an arc having x as initial point and y as terminal point.

A pathwise connected space is always a connected space, but a connected space need not be path connected. An arcwise connected space is always a pathwise connected space, but a pathwise connected space need not be arcwise connected.As it turns out, for Hausdorff spaces these two notions coincide — a Hausdorff space is pathwise connected iff it is arcwise connected.

Titlepath
Canonical namePath
Date of creation2013-03-22 12:00:15
Last modified on2013-03-22 12:00:15
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id15
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 54D05
Synonympathwise connected
Synonympath-connected
Synonympath connected
Related topicSimplePath
Related topicDistanceInAGraph
Related topicLocallyConnected
Related topicExampleOfAConnectedSpaceWhichIsNotPathConnected
Related topicPathConnectnessAsAHomotopyInvariant
Definespath
Definesarc
Definesarcwise connected
Definesinitial point
Definesterminal point
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更新时间:2025/5/4 6:06:01