Path (topology)
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Important page, there are other kinds of path. Need to be mentioned
- Note: see Path for other uses of the term "path".
Contents
Definition
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
Notes
- ↑ Sometimes denoted [ilmath]I[/ilmath]
- ↑ See also: definitions and iff