ε ≥ 0 A ⊆ A[ε], Ø[ε] = Ø, (A ∪ B)[ε] = A[ε] ∪ B[ε], A[ε][δ] ⊆ A[ε+δ], A[ε] = ∩δ>ε A[δ]. Given a distance d, the associated A → A(ε) is a tower. Conversely, given a tower, the map d(x,A) = inf{ε : x ∈ A[ε]} is a distance, and these two operations are inverses of each other. A contraction f:(X, d)→(Y, e) is, in terms of associated towers, a map such that for all ε ≥ 0, f[A[ε]] ⊆ f[A][ε]. Categorical properties The main interest in approach spaces and their contractions is that they form a category with good properties, while still being quantitative like metric spaces. One can take arbitrary products, coproducts, and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like βN, the Stone–Čech compactification of the integers. Certain hyperspaces, measure spaces, and probabilistic metric spaces turn out to be naturally endowed with a distance. Applications have also been made to approximation theory. References External links Robert Lowen Closure operators