# path

Let $I=[0,1]\subset\mathbb{R}$ and let $X$ be a topological space.

A continuous map $f:I\rightarrow X$ 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 equivalent if they differ by a reparameterization. That is to say, we regard $f\colon I\to X$ and $g\colon I\to X$ as equivalent if there exists a homeomorphism $h\colon I\to I$ such that $h(0)=0$ and $h(1)=1$ and $f=g\circ h$.

If the space $X$ has extra structure, 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 differentiable paths. Likewise, one can speak of piecewise linear paths, rectifiable paths, and analytic paths in suitable contexts.

The space $X$ is said to be pathwise connected if, for every two points $x,y\in X$, 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 $x,y\in X$, 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