Difference between revisions of "Loop (topology)"

Latest revision as of 20:32, 1 November 2016

Note: see loop for other uses of the term.

Definition

Let $p:[0,1]\rightarrow X$ be a path exactly as is defined on that page, then^[1]:

• $p$ is a loop if^{[Note 1]}:
  • $p(0)=p(1)$, or in words: the initial point equals the terminal point of the path.

We call $p(0)=p(1)=x_0$ the base point of the loop.

The constant loop based at $x_0\in X$ is the loop: $\ell:[0,1]\rightarrow X$ given by $\ell:t\mapsto x_0$

See also

• Loop concatenation - creating a new loop, $\ell_1*\ell_2$ from loops with the same basepoint, $\ell_1$ and $\ell_2$.
• Path and loop. These are always related no matter the context.
  • Path (topology)
• Constant loop based at a point (AKA: constant loop)
• First homotopy group, $\pi_1(X,x_0)$ - a group structure defined on equivalence classes of loops in a topological space, $(X,\mathcal{ J })$, based at $x_0$

Notes

1. Jump up ↑ See also: Definitions and iff

References

1. Jump up ↑ Introduction to Topological Manifolds - John M. Lee