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

Latest revision as of 21:03, 1 November 2016

Definition

Let $(X,\mathcal{ J })$ be a topological space, and let $b\in X$ be given ("the point" in the title). There is a "special" loop called "the constant loop based at $b$", say $\ell:I\rightarrow X$^{[Note 1]} such that:

• $\ell:t\mapsto b$.
  • Yes, a constant map: $\forall t\in I[\ell(t)=b]$.
  • Claim 1: this is a loop based at $b$

It is customary (and a convention we almost always use) to write a constant loop based at $b$ as simply: $b$.

This is really a special case of a constant map.

Terminology and synonyms

Terminology

We use $b:I\rightarrow X$ (or just "let $b$ denote the constant loop based at $b\in X$") for a few reasons:

1. Loop concatenation of $\ell_1:I\rightarrow X$ and $b$ (where $\ell_1$ is based at $b$) can be written as:
   • $\ell_1*b$
2. In the context of path homotopy classes, we will write things like $[\ell_1][b]=[\ell_1*b]=[\ell_1]$, this is where the notation really becomes useful and plays very nicely with Greek or not-standard letters (like $\ell$ rather than $l$) for non-constant loops.

(See: The fundamental group for details)

Synonyms

Other names include:

1. Trivial loop
2. Trivial loop based at a point
3. Trivial loop based at $b\in x$

Proof of claims

Writing $b(0)=b$ is very confusing, so here we denote by $\ell$ the constant loop based at $b\in X$.

Claim 1: $\ell$ is a loop based at $b$

There are two parts to prove:

1. $\ell$ is continuous, and
2. $\ell$ is based at $b$

We consider $I$ 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 $\ell:I\rightarrow X$.
   • Let $U\in\mathcal{J}$ be given (so $U$ is an open set in $(X,\mathcal{ J })$)
     • If $b\in U$ then $\ell^{-1}(U)=I$ which is open in $I$
     • If $b\notin U$ then $\ell^{-1}(U)=\emptyset$ which is also open in $I$.
2. That $\ell$ is a loop based at $b\in X$:
   • As $\forall t\in I[\ell(t)=b]$ we see in particular that:
     1. $\ell(0)=b$ and
     2. $\ell(1)=b$

See also

• The fundamental group
• Loop
• Loop concatenation

Notes

1. Jump up ↑ Where $I:=[0,1]\subset\mathbb{R}$ (the unit interval)

References