Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem (Q719089)

There are various ways in which one can characterize covering dimension; one such way is by counting the number of open sets of a certain diameter needed to cover the whole space. E.g., the unit interval can be covered by roughly \(n\) open sets of diameter~\(1/n\), whereas the unit square requires roughly \(n^2\) such sets. This suggests that the quotient of the logarithm of the number of sets and the logarithm of the diameter may be related to the dimension of the space and, indeed, the Pontrjagin-Schnirelmann theorem gives an asymptotic formula for the covering dimension of a compact metrizable space that involves all possible diameters and all compatible metrics on the space. The authors derive similar formulas for the dimension of separable metrizable spaces using normal sequences of finite open covers. Given a sequence \(U=\langle\mathcal{U}_i:i\in\mathbb{N}\rangle\) of open covers one defines two numbers \(d_3(U)=\liminf_{i\to\infty}{1\over i}\log_3|\mathcal{U}_i|\) and \(d_2(U)=\liminf_{i\to\infty}{1\over i}\log_2|\mathcal{U}_i|\). The authors establish that \(\dim X\) is equal to the minimum value of \(d_3(U)\) taken over all sequences~\(U\) that are developments and for which \(\mathcal{U}_{i+1}\) is always a star-refinement of~\(\mathcal{U}_i\); a similar result with \(d_2(U)\) is obtained when one considers developments where \(\mathcal{U}_{i+1}\)~is always a barycentric refinement of~\(\mathcal{U}_i\).