An 
-space, named after Heinz Hopf, and sometimes
also called a Hopf space, is a topological space
together with a continuous binary operation 
, such that there exists a point 
with the property that the two maps 
and 
are both homotopic to the identity map 
on 
,
through homotopies preserving the point 
. The element 
is called a homotopy identity.
One should note that authors do not always agree on the definition of an 
-space. In some texts, the maps given by 
and 
are required to be equal to the identity on 
. In others, the two maps are required to be homotopic to the
identity as above, but the homotopies need not fix the element 
. Fortunately, we have the comforting fact that for any CW-complex, the three definitions above are equivalent.
For any 
-space 
with homotopy identity 
, the fundamental group
with base-point 
is an Abelian group. Taking another base-point 
in a path-component of 
not containing 
may, however, result in a non-Abelian fundamental group.
A deep theorem in homotopy theory known as the Hopf invariant one theorem (sometimes also known as Adams' theorem) states that the
only 
-spheres that are 
-spaces are 
, 
,
, and 
.
