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Homeloop

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# loop

A loop based at $x_{0}$ in a topological space $X$ is simply a continuous map $f:[0,1]\to X$ with $f(0)=f(1)=x_{0}$.

The collection of all such loops, modulo homotopy equivalence, forms a group known as the fundamental group.

More generally, the space of loops in $X$ based at $x_{0}$ with the compact-open topology, represented by $\Omega_{{x_{0}}}$, is known as the loop space of $X$. And one has the homotopy groups $\pi_{n}(X,x_{0})=\pi_{{n-1}}(\Omega_{{x_{0}}},\iota)$, where $\pi_{n}$ represents the higher homotopy groups, and $\iota$ is the basepoint in $\Omega_{{x_{0}}}$ consisting of the constant loop at $x_{0}$.

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Reference

## Mathematics Subject Classification

54-00*no label found*

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