Path (topology)

From Maths
Jump to: navigation, search
Stub grade: B
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Important page, there are other kinds of path. Need to be mentioned
Note: see Path for other uses of the term "path".


Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath][0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}[/ilmath] denote the closed unit interval[Note 1], considered as a topological subspace of [ilmath]\mathbb{R} [/ilmath] with its usual topology, and let [ilmath]p:[0,1]\rightarrow X[/ilmath] be a map. Then[1]:

  • [ilmath]p[/ilmath] is called a path if [ilmath]p[/ilmath] is continuous[Note 2]
    • [ilmath]p(0)[/ilmath] is the initial point of the path
    • [ilmath]p(1)[/ilmath] is the terminal point of the path

Note: path usually means a curve on a bounded and connected subspace of [ilmath]\mathbb{R} [/ilmath], say [ilmath]A[/ilmath], so [ilmath]p:A\rightarrow X[/ilmath] is a path. It need not be [ilmath][0,1][/ilmath]. The context will always make this clear.
Note: paths in other contexts may require additional properties, eg smoothness, differentiability, so forth

See also

  • Loop (topology) - a path where the start and end points are the same, that is for [ilmath]p:[0,1]\rightarrow X[/ilmath] we have [ilmath]p(0)=p(1)[/ilmath]
  • Path homotopy


  1. Sometimes denoted [ilmath]I[/ilmath]
  2. See also: definitions and iff


  1. Introduction to Topological Manifolds - John M. Lee