Difference between revisions of "Omega(X,b)"
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These are clearly different {{plural|function|s}} | These are clearly different {{plural|function|s}} | ||
| + | ==See also== | ||
| + | * [[The set of continuous functions between topological spaces]] | ||
| + | * [[C(I,X)|{{M|C(I,X)}}]] | ||
| + | * [[Paths and loops in a topological space]] | ||
| + | * [[The fundamental group]] | ||
| + | * [[Index of spaces, sets and classes]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Algebraic Topology|Homotopy Theory|Topology|Functional Analysis}} | {{Definition|Algebraic Topology|Homotopy Theory|Topology|Functional Analysis}} | ||
Latest revision as of 04:47, 3 November 2016
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]b\in X[/ilmath] be given. Then [ilmath]\Omega(X,b)\subseteq[/ilmath][ilmath]C([0,1],X)[/ilmath] is the set containing all loops based at [ilmath]b[/ilmath][1]. That is:
- [ilmath](\ell:I\rightarrow X)\in\Omega(X,b)[/ilmath] means [ilmath]\ell[/ilmath] is a loop based at [ilmath]b\in X[/ilmath] (AKA: a path such that [ilmath]\ell(0)=\ell(1)=b[/ilmath])
There is additional structure we can imbue on [ilmath]\Omega(X,b)[/ilmath]:
- [ilmath]*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)[/ilmath] - the operation of loop concatenation:
- [ilmath]*:(\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)[/ilmath]
Caution:This is not a monoid or even a semigroup as [ilmath]*[/ilmath] 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 [ilmath]\alpha,\beta,\gamma\in\Omega(X,b)[/ilmath] that [ilmath]\alpha*(\beta*\gamma)\ne(\alpha*\beta)*\gamma[/ilmath], that is because [ilmath]\alpha*(\beta*\gamma)[/ilmath] spends [ilmath]0\le t\le \frac{1}{2} [/ilmath] doing [ilmath]\alpha[/ilmath] at double speed, then does [ilmath]\beta[/ilmath] during [ilmath]\frac{1}{2}\le t\le \frac{3}{4} [/ilmath] at 4x the normal speed, then lastly [ilmath]\gamma[/ilmath] during [ilmath]\frac{3}{4}\le t\le 1[/ilmath] at 4x the normal speed also.
In contrast, [ilmath](\alpha*\beta)*\gamma[/ilmath] does [ilmath]\alpha[/ilmath] at 4x normal speed during [ilmath]0\le t\le \frac{1}{4}[/ilmath] then [ilmath]\beta[/ilmath] at 4x normal speed during [ilmath]\frac{1}{4}\le t\le \frac{1}{2} [/ilmath] then lastly, [ilmath]\gamma[/ilmath] at double speed during [ilmath]\frac{1}{2}\le t\le 1[/ilmath]
These are clearly different functions
See also
- The set of continuous functions between topological spaces
- [ilmath]C(I,X)[/ilmath]
- Paths and loops in a topological space
- The fundamental group
- Index of spaces, sets and classes
