Definition 2

A pointed topological space $(X, e)$ endowed with a continuous law of composition $m: X\times X\rightarrow X$ is called a Hopf space, such that

(i) $m(e, e) =e$;

(ii) there exist homotopies pointed at $e$ connecting the mappings $x\mapsto m(x, e)$ and $x\mapsto m(e, x)$ with the mapping Id$_X$.

Properties (i) and (ii) are sometimes expressed by saying that $e$ is an identity element up to homotopy for the composition law $m$.

It should be noted that there may exist several identity elements up to homotopy for a continuous composition law $m$ on a topological space X. For example, for $X =\mathbf{R}$ and $m(x, y) = (x+y)/2$, every $x\in \mathbf{R}$ is an identity element up to homotopy.