In this paper we obtain a modified version of the classical theorem of Pontrjagin
and Schnirelmann concerning the covering dimension of a compact metric space and
the box-counting dimensions associated with the metrics on the space. T. Miyata and T.
Watanabe defined box-counting dimension for approximate sequences X which represent
compact metric spaces X as approximate resolutions p : X → X. We show that the
covering dimension equals the infimum of the box-counting dimensions associated with
the approximate resolutions of the space.