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.