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| A {\em loop} based at $x_0$ in a topological space $X$ is simply a continuous map $f : [0,1]\to X$ with $f(0) = f(1) = x_0$. |
A {\em loop} based at $x_0$ in a topological space $X$ is simply a continuous map $f : [0,1]\to X$ with $f(0) = f(1) = x_0$. |
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| The collection of all such loops, modulo homotopy equivalence, forms a group known as the fundamental group. |
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| 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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