We present simple characterizations of the sets Eμ and Ex of measure entropy pairs and topological entropy pairs of a topological dynamical system (X,T) with invariant probability measure μ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relation P forms a residual subset of the set EX. Finally an example of a minimal point distal dynamical system is constructed for which EX ∩ (X0 × X0) ≠ ∅, where X0 is the dense Gδ subset of distal points in X.