Difference between revisions of "Loop (topology)"
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The ''constant loop based at {{M|x_0\in X}}'' is the loop: {{M|\ell:[0,1]\rightarrow X}} given by {{M|\ell:t\mapsto x_0}} | The ''constant loop based at {{M|x_0\in X}}'' is the loop: {{M|\ell:[0,1]\rightarrow X}} given by {{M|\ell:t\mapsto x_0}} | ||
==See also== | ==See also== | ||
| + | * [[Loop concatenation]] - creating a new loop, {{M|\ell_1*\ell_2}} from loops with the same basepoint, {{M|\ell_1}} and {{M|\ell_2}}. | ||
* [[Path]] and [[loop]]. These are always related no matter the context. | * [[Path]] and [[loop]]. These are always related no matter the context. | ||
** [[Path (topology)]] | ** [[Path (topology)]] | ||
* [[Constant loop based at a point]] ({{AKA}}: [[constant loop]]) | * [[Constant loop based at a point]] ({{AKA}}: [[constant loop]]) | ||
* [[First homotopy group]], {{M|\pi_1(X,x_0)}} - a [[group]] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}} | * [[First homotopy group]], {{M|\pi_1(X,x_0)}} - a [[group]] structure defined on [[equivalence classes]] of loops in a [[topological space]], {{Top.|X|J}}, based at {{M|x_0}} | ||
| + | |||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 20:32, 1 November 2016
Stub grade: A
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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:
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
