Scientific publications

We define the topological dimension dim ξ of maximal linked systems (m.l.s.) of closed subsets of a metrizable compact space X. By definition, dim ξ is the greatest lower bound of the lower quantization dimensions of the m.l.s. ξ over all metrics compatible with the topology of X. The proposed definition of dim ξ is motivated by the Pontryagin – Shnirelman theorem, which characterizes the topological dimension of a metrizable compact space as the infimum of the lower box dimensions of this compactum over all compatible metrics. For any metrizable compact space X and any m.l.s. ξ, the inequality dim ξ dimX holds. Moreover, for a finitedimensional metrizable compactum X, the following theorem on intermediate values of the dimension dim ξ holds: for any integer k satisfying the inequalities 0 k dimX, there exists a m.l.s. ξk such that dim ξk = k. It is proved that for all previously considered maximal linked systems with known values of the quantization dimensions, the dimension dim ξ is an integer. In particular, m.l.s. ξ(A,B) constructed using E. V. Kashuba’s technology are always zero-dimensional in the topological sense.