Axiomatic method of partitions in the theory of Noebeling spaces. I. Improvement of partition connectivity

The Noebeling space Nk2k+1, a k-dimensional analogue of the Hilbert space, is considered; this is a topologically complete separable (that is, Polish) k-dimensional absolute extensor in dimension k (that is, AE(k)) and a strongly k-universal space. The conjecture that the above-listed properties characterize the Noebeling space Nk2k+1 in an arbitrary finite dimension k is proved. In the first part of the paper a full axiom system of the Noebeling spaces is presented and the problem of the improvement of a partition connectivity is solved on its basis. Bibliography: 29 titles.