$\Omega(X,b)$

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Definition

Let $(X,\mathcal{ J })$ be a topological space and let $b\in X$ be given. Then $\Omega(X,b)\subseteq$$C([0,1],X)$ is the set containing all loops based at $b$. That is:

• $(\ell:I\rightarrow X)\in\Omega(X,b)$ means $\ell$ is a loop based at $b\in X$ (AKA: a path such that $\ell(0)=\ell(1)=b$)

There is additional structure we can imbue on $\Omega(X,b)$:

• $*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)$ - the operation of loop concatenation:
  • $*:(\ell_1,\ell_2)\mapsto\left((\ell_1*\ell_2):[0,1]\rightarrow X\text{ by }(\ell_1*\ell_2):t\mapsto\left\{ \begin{array}{lr} \ell_1(2t) & \text{for }t\in[0,\frac{1}{2}] \\ \ell_2(2t-1) & \text{for }t\in[\frac{1}{2},1] \end{array} \right.\right)$

Caution:This is not a monoid or even a semigroup as $*$ is not associative. See "Caveats" below

This set and the operation of loop concatenation are a precursor for the fundamental group

Caveats

Associativity (or lack of)

Note that for $\alpha,\beta,\gamma\in\Omega(X,b)$ that $\alpha*(\beta*\gamma)\ne(\alpha*\beta)*\gamma$, that is because $\alpha*(\beta*\gamma)$ spends $0\le t\le \frac{1}{2}$ doing $\alpha$ at double speed, then does $\beta$ during $\frac{1}{2}\le t\le \frac{3}{4}$ at 4x the normal speed, then lastly $\gamma$ during $\frac{3}{4}\le t\le 1$ at 4x the normal speed also.

In contrast, $(\alpha*\beta)*\gamma$ does $\alpha$ at 4x normal speed during $0\le t\le \frac{1}{4}$ then $\beta$ at 4x normal speed during $\frac{1}{4}\le t\le \frac{1}{2}$ then lastly, $\gamma$ at double speed during $\frac{1}{2}\le t\le 1$

These are clearly different functions

See also

• The set of continuous functions between topological spaces
• $C(I,X)$
• Paths and loops in a topological space
• The fundamental group
• Index of spaces, sets and classes