Loop (topology)
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Created as a stub. Be sure to create disambiguation page loop.
Tasks:
- Don't be lazy, give full definition
- Note: see loop for other uses of the term.
Contents
Definition
Let [ilmath]p:[0,1]\rightarrow X[/ilmath] be a path exactly as is defined on that page, then^{[1]}:
- [ilmath]p[/ilmath] is a loop if^{[Note 1]}:
- [ilmath]p(0)=p(1)[/ilmath], or in words: the initial point equals the terminal point of the path.
We call [ilmath]p(0)=p(1)=x_0[/ilmath] the base point of the loop.
The constant loop based at [ilmath]x_0\in X[/ilmath] is the loop: [ilmath]\ell:[0,1]\rightarrow X[/ilmath] given by [ilmath]\ell:t\mapsto x_0[/ilmath]
See also
- Loop concatenation - creating a new loop, [ilmath]\ell_1*\ell_2[/ilmath] from loops with the same basepoint, [ilmath]\ell_1[/ilmath] and [ilmath]\ell_2[/ilmath].
- Path and loop. These are always related no matter the context.
- Constant loop based at a point (AKA: constant loop)
- First homotopy group, [ilmath]\pi_1(X,x_0)[/ilmath] - a group structure defined on equivalence classes of loops in a topological space, [ilmath](X,\mathcal{ J })[/ilmath], based at [ilmath]x_0[/ilmath]
Notes
- ↑ See also: Definitions and iff