Supplement to the Pontryagin–Schnirelmann theorem

The lower box dimension $\underline{\dim}_B$ of a metric compactum $(X,\rho)$ appeared originally in 1932 in the work of Pontryagin and Schnirelmann, who proved that $\underline{\dim}_BX$ is always greater than or equal to the topological dimension $\dim X$ and each metrizable compactum admits a metric with $\underline{\dim}_BX=\dim X$. The present article shows that, given an infinite metrizable compactum $X$ and a real $b$ satisfying $\dim X\leq b\leq\infty$, there exists a metric on $X$ compatible with the topology such that $\underline{\dim}_BX=b$.