I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283):

**Definition**: A *Hopf algebra* is a graded algebra $A = \oplus_{n \geq 0} A^n$ over a commutative base ring $R$ satisfying the following two conditions:

- There is an identity element $1 \in A^0$ such that the map $R \rightarrow A^0$, $r \mapsto r \cdot 1$, is an isomorphism; one says that $A$ is
*connected*. - There is a
*diagonal*or*coproduct*$\Delta\colon A \rightarrow A \otimes A$, a homomorphism of graded algebras satisfying $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + \sum_{0<i<n} \alpha'_i \otimes \alpha''_{n-i}$ for $\alpha \in A^n$, $n >0$, and $\alpha'_j,\alpha''_j \in A^j$.

Of course, in other settings Hopf algebras don't need to be graded, but I guess graded ones are the ones he cares about. Here are my two questions:

In condition 2, one could instead ask for the a priori weaker condition that $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + x$ with $x \in \oplus_{0<i<n} A^i \otimes A^{n-i}$. It seems to me that in his examples (all coming from H-spaces) this is all that holds, at least naively. Is this a mistake, or is this really the right definition?

When I look at definitions of Hopf algebras elsewhere, there is also an

*antipode*map. How can you derive this in Hatcher's setup?