Constant loop based at a point
Contents
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, and let [ilmath]b\in X[/ilmath] be given ("the point" in the title). There is a "special" loop called "the constant loop based at [ilmath]b[/ilmath]", say [ilmath]\ell:I\rightarrow X[/ilmath]^{[Note 1]} such that:
- [ilmath]\ell:t\mapsto b[/ilmath].
- Yes, a constant map: [ilmath]\forall t\in I[\ell(t)=b][/ilmath].
- Claim 1: this is a loop based at [ilmath]b[/ilmath]
It is customary (and a convention we almost always use) to write a constant loop based at [ilmath]b[/ilmath] as simply: [ilmath]b[/ilmath].
This is really a special case of a constant map.
Terminology and synonyms
Terminology
We use [ilmath]b:I\rightarrow X[/ilmath] (or just "let [ilmath]b[/ilmath] denote the constant loop based at [ilmath]b\in X[/ilmath]") for a few reasons:
- Loop concatenation of [ilmath]\ell_1:I\rightarrow X[/ilmath] and [ilmath]b[/ilmath] (where [ilmath]\ell_1[/ilmath] is based at [ilmath]b[/ilmath]) can be written as:
- [ilmath]\ell_1*b[/ilmath]
- In the context of path homotopy classes, we will write things like [ilmath][\ell_1][b]=[\ell_1*b]=[\ell_1][/ilmath], this is where the notation really becomes useful and plays very nicely with Greek or not-standard letters (like [ilmath]\ell[/ilmath] rather than [ilmath]l[/ilmath]) for non-constant loops.
(See: The fundamental group for details)
Synonyms
Other names include:
- Trivial loop
- Trivial loop based at a point
- Trivial loop based at [ilmath]b\in x[/ilmath]
Proof of claims
Writing [ilmath]b(0)=b[/ilmath] is very confusing, so here we denote by [ilmath]\ell[/ilmath] the constant loop based at [ilmath]b\in X[/ilmath].
Claim 1: [ilmath]\ell[/ilmath] is a loop based at [ilmath]b[/ilmath]
There are two parts to prove:
- [ilmath]\ell[/ilmath] is continuous, and
- [ilmath]\ell[/ilmath] is based at [ilmath]b[/ilmath]
We consider [ilmath]I[/ilmath] with the topology it inherits from the usual topology of the reals. That is the topology induced by the absolute value as a metric.
Proof
- Continuity of [ilmath]\ell:I\rightarrow X[/ilmath].
- Let [ilmath]U\in\mathcal{J} [/ilmath] be given (so [ilmath]U[/ilmath] is an open set in [ilmath](X,\mathcal{ J })[/ilmath])
- If [ilmath]b\in U[/ilmath] then [ilmath]\ell^{-1}(U)=I[/ilmath] which is open in [ilmath]I[/ilmath]
- If [ilmath]b\notin U[/ilmath] then [ilmath]\ell^{-1}(U)=\emptyset[/ilmath] which is also open in [ilmath]I[/ilmath].
- Let [ilmath]U\in\mathcal{J} [/ilmath] be given (so [ilmath]U[/ilmath] is an open set in [ilmath](X,\mathcal{ J })[/ilmath])
- That [ilmath]\ell[/ilmath] is a loop based at [ilmath]b\in X[/ilmath]:
- As [ilmath]\forall t\in I[\ell(t)=b][/ilmath] we see in particular that:
- [ilmath]\ell(0)=b[/ilmath] and
- [ilmath]\ell(1)=b[/ilmath]
- As [ilmath]\forall t\in I[\ell(t)=b][/ilmath] we see in particular that:
See also
Notes
- ↑ Where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] (the unit interval)
References
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