Axiomatic method of partitions in the theory of N¨obeling spaces. I. Improvement of partition connectivity

Abstract

The N¨obeling space N 2k+1 k , 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 N¨obeling space N 2k+1 k in an arbitrary finite dimension k is proved. In the first part of the paper a full axiom system of the N¨obeling spaces is presented and the problem of the improvement of a partition connectivity is solved on its basis.