View source for Omega(X,b)

{{DISPLAYTITLE:{{M|\Omega(X,b)}}}}{{Stub page|grade=B|msg=Check, maybe find another reference, remove as needed}}
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==Definition==
Let {{Top.|X|J}} be a [[topological space]] and let {{M|b\in X}} be given. Then {{M|\Omega(X,b)\subseteq}}[[C(I,X)|{{M|C([0,1],X)}}]] is the set containing all [[loops]] based at {{M|b}}{{rITTMJML}}. That is:
* {{M|(\ell:I\rightarrow X)\in\Omega(X,b)}} means {{M|\ell}} is a [[loop]] based at {{M|b\in X}} ({{AKA}}: a [[path]] such that {{M|1=\ell(0)=\ell(1)=b}})
There is additional structure we can imbue on {{M|\Omega(X,b)}}:
* {{M|*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)}} - the operation of [[loop concatenation]]:
** {{M|*:(\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 {{M|*}} is not [[associative]]. See [[#Caveats|"''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 {{M|\alpha,\beta,\gamma\in\Omega(X,b)}} that {{M|\alpha*(\beta*\gamma)\ne(\alpha*\beta)*\gamma}}, that is because {{M|\alpha*(\beta*\gamma)}} spends {{M|0\le t\le \frac{1}{2} }} doing {{M|\alpha}} at double speed, then does {{M|1=\beta}} during {{M|\frac{1}{2}\le t\le \frac{3}{4} }} at 4x the normal speed, then lastly {{M|\gamma}} during {{M|\frac{3}{4}\le t\le 1}} at 4x the normal speed also.

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

These are clearly different {{plural|function|s}}
==References==
<references/>
{{Definition|Algebraic Topology|Homotopy Theory|Topology|Functional Analysis}}