James construction

Definition

The James construction is a functor from based topological spaces to topological monoids, as follows. Given a topological space $X$, $JX$ is the monoid whose elements are words with letters coming from $X$, modulo the relation that the basepoint in $X$ is a two-sided identity element.

$JX$ has a natural filtration where the $n^{th}$ component is the set of elements which can be expressed as words of length at most $n$. Equip the $n^{th}$ filtered component with the quotient topology from $X^{n}$, and define the topology on $JX$ as the union topology from the filtered components.