Path (topology)

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".

Definition

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

• $p$ is called a path if $p$ is continuous
  • $p(0)$ is the initial point of the path
  • $p(1)$ is the terminal point of the path

Note: path usually means a curve on a bounded and connected subspace of $\mathbb{R}$, say $A$, so $p:A\rightarrow X$ is a path. It need not be $[0,1]$. 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 $p:[0,1]\rightarrow X$ we have $p(0)=p(1)$
• Path homotopy