Let T : X → X be a continuous selfmap of a compact metrizable space X. We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X. (2) There is a countable point separating family F ⊂ C(X) of non-negative functions f ∈ C(X) such that for every f ∈ F there is g ∈ C(X) with f = g − g ◦ T.
Pointfree formulas for three kinds of separating points for closed sets by maps are given. These formulas allow controlling the amount of factors of the target product space so that it does not exceed the weight of the embeddable space. In literature, the question of how many factors of the target product are needed for the embedding has only been considered for specific spaces. Our approach is algebraic in character and can thus be viewed as a contribution to Kuratowski's topological calculus.