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.

Title path
Canonical name Path
Date of creation 2013-03-22 12:00:15
Last modified on 2013-03-22 12:00:15
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 15
Author rspuzio (6075)
Entry type Definition
Classification msc 54D05
Synonym pathwise connected
Synonym path-connected
Synonym path connected
Related topic SimplePath
Related topic DistanceInAGraph
Related topic LocallyConnected
Related topic ExampleOfAConnectedSpaceWhichIsNotPathConnected
Related topic PathConnectnessAsAHomotopyInvariant
Defines path
Defines arc
Defines arcwise connected
Defines initial point
Defines terminal point