Factorization and Compactification Theorems for Separable Metric Spaces

Abstract

Recall that in a metric space (X, d), the diameter of a subset A of X is defined by $\mathop {\mathrm {diam}} \nolimits (A) =\sup \{d(x, y): x, y\in A\}$, and the mesh of a collection $\mathcal {G}$ of subsets of X by $\mathop {\mathrm {mesh}} \nolimits (\mathcal {G})= \sup \{ \mathop {\mathrm {diam}} \nolimits (G): G \in \mathcal {G}\}$. In both cases the supremum is taken in the set of non-negative real numbers, so that $\mathop {\mathrm {diam}} \nolimits (\emptyset )= \mathop {\mathrm {mesh}} \nolimits (\emptyset ) = 0$.