A compact quantum group, $Q_{{CG}}$ is defined as a particular case of a locally compact quantum group $Q_{{LCG}}$, that is, a quadruple $(A,\Delta,\mu,\nu)$, where $A$ is either a $C^{*}$– or a $W^{*}$– algebra equipped with a co-associative comultiplication $\Delta:A\to A\otimes A$, and two faithful semi-finite normal weights, $\mu$ and $\nu$ –right and -left Haar measures, and also when the object space $\mathbf{O}$ of the latter $Q_{{LCG}}$ is replaced by a compact topological space $Q^{T}_{{CG}}$, instead of being a locally compact topological space like $Q_{{LCG}}$.