Difference between revisions of "Constant loop based at a point"

Revision as of 20:54, 1 November 2016

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:

1. 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]
2. 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:

1. Trivial loop
2. Trivial loop at a point
3. 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:

1. [ilmath]\ell[/ilmath] is continuous, and
2. [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

1. 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].
2. 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:
     1. [ilmath]\ell(0)=b[/ilmath] and
     2. [ilmath]\ell(1)=b[/ilmath]

See also

• The fundamental group
• Loop
• Loop concatenation

Notes

1. ↑ Where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] (the unit interval

References