H-group

Definition
A H-group is a set $$G$$ equipped with the structure of a topological space and three operations:

satisfying two kinds of compatibility conditions -- continuity conditions and homotopy versions of group law conditions.

The continuity conditions are as follows:

The homotopy version of group law conditions are as follows:

It's important to note here that unlike the ordinary associativity, identity element, and inverse element conditions, the homotopy versions cannot be checked separately for each tuple of elements. Rather, to check for the truth of the homotopy version, we need to look at the mappings (such as $$(a,b,c) \mapsto (a * b) * c$$) in their entirety as we try to homotope them.

Contractible spaces
If $$G$$ is equipped with a topology that makes it a contractible space, then the homotopy versions of the group laws are vacuously satisfied, because all maps are nullhomotopic. In other words, in this case, we can select any continuous choice for the multiplication, identity element, and inverse map.