Infinite-dimensional compactum without intermediate values of the lower box dimension

The term «(bi)compactum without intermediate dimensions» was introduced by V. V. Fedorchuk in 1973 to denote compact spaces of topological (Lebesgue) dimension n, all non-empty closed subsets of which are either zero-dimensional or also have dimension n. The box dimensions (upper dimB and lower dimB) of metric compacta can take any non-negative value (including infinity), and the question of intermediate values of their dimension is formulated as follows. Let a metric compact space X have a box dimension (upper or lower) equal to a. Is it true that for any non-negative b < a there exists a closed subset in X whose corresponding box dimension is equal to b? The answer for the upper box dimension is known to be positive. For the lower box dimension, the author has previously constructed an example of a one-dimensional (in the sense of dimB) metric compactum without intermediate values of dimB. In this paper, we prove the following theorem, which strengthens this result: for any positive real number a ∞ there exists a metric compact space X of dimension dimB X = a, in which all non-empty proper closed subsets have the dimension dimB equal to zero. Thus, there exist metric compacta of any predefined box dimension a without intermediate values of the lower box dimension. The case of greatest interest here is a = ∞.