Space of path components
As a set and topological space
- As a set, it is the set of path components of . In other words, it is the set of equivalence classes of under the equivalence relation iff there is a path from to in .
- As a topological space, it is obtained by taking the quotient topology of under the equivalence relation iff there is a path from to in .
As a monoid
If is a topological group, topological monoid, or (most generally) a H-space, then naturally acquires the structure of a monoid (in fact, a topological monoid, but a discrete one if is locally path-connected). The structure is obtained by quotienting the original monoid (or H-space multiplication) by the equivalence relation of being path-connected.
In order to perform this quotienting, we need to justify that the relation of being in the same path component is a congruence for the original monoid (or H-space multiplication). This is indeed guaranteed by the continuity of multiplication: if and are in the same path component, and and are in the same path component, then and are in the same path component, because the paths can be multiplied pointwise.
Further, even if the original multiplication was not associative but only associative up to homotopy (making it an H-space), the new multiplication is strictly associative because any homotopy must descend to the identity map at the level of path components.