From a compact topological group to a commutative Hopf algebra

Question

As we know we can associate a commutative Hopf algebra to any compact topological group as follows:

Let $G$ be a compact topological group. Consider the space of continuous functions on $G$ denoted by $C(G)$ together with the following maps:

$(f.h)(g) = f(g)h(g)$, $~~~$ $\Delta(f)(g_1\otimes g_2) = f(g_1g_2)$, $~~~$

$\eta(x) = x 1$ where $1(g)=1$ for all $g\in G$,

$\epsilon(f) = f(e)$ where $e$ is the unit element of $G$,

$S(f)(g) = f(g^{−1})$.

Now my question is that why to be compact is important here for our topological group?

Thank you for your help!