Loop (topology)

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Note: see loop for other uses of the term.

Definition

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

• [ilmath]p[/ilmath] is a loop if:
  • [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.
  • Path (topology)
• 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]