# path

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

A continuous map^{} $f:I\to X$ 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 $f:I\to X$ and $g:I\to X$ as equivalent if there exists a homeomorphism $h:I\to I$ 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 |