Axiomatic method of partitions in the theory of Nöbeling spaces. I: Improvement of partition connectivity. (English. Russian original) Zbl 1147.54019

Ageev, S. M.

Sb. Math. 198, No. 3, 299-342 (2007); translation from Mat. Sb. 198, No. 3, 3-50 (2007).

This is the first part of a series of three papers in which the author proves a classical conjecture on a characterization of Nöbeling spaces.
For \(Y \subseteq X\) and \(0 \leq k \leq n \leq \infty\), let
\[ \Sigma^{n}_{k}(X;Y) = \{(x_{i})_{i < n} \in X^{n}: |\{i < n : x_{i} \in X \setminus Y\}| \leq k\}. \] Let \(\mathcal J \subseteq \mathbb R\) denote the set of irrational numbers. Then the Nöbeling spaces are \(N_{k}^{2k+1}=\Sigma_{k}^{2k+1}(\mathbb R, \mathcal J)\) where \(0 \leq k \leq \infty\). It turns out that for every \(0 \leq k \leq \infty\),

(1)

\(N_{k}^{2k+1}\) is a Polish space of topological dimension \(k\);

(2)

\(N_{k}^{2k+1}\) is an absolute extensor in dimension \(k\), that is for every separable metric space \(Z\) with \(\text{dim }Z \leq k\) and any closed set \(A \subseteq Z\), every continuous function \(f : A \to X\) can be extended to a continuous function \(\widehat f : Z \to X\) with \(\widehat f |_{A} = f\);

(3)

\(N_{k}^{2k+1}\) is a strongly \(k\)-universal Polish space, that is for every Polish space \(Z\) with \(\text{dim }Z \leq k\), every continuous function \( f : Z \to X\) can be approximated arbitrarily closely by a closed embedding.

According to classical results of P. S. Alexandroff and P. S. Urysohn [Math. Ann. 98, 89–106 (1927; JFM 53.0559.01)], H. Toruńczyk [Fundam. Math. 111, 247–262 (1981; Zbl 0468.57015)] and K. Kawamura, M. Levin and E. D. Tymchatyn [Topol. Proc. 22, 155–174 (1997; Zbl 0945.54029)], in the \(k=0\), \(k=\infty\) and \(k=1\) special cases properties (1)–(3) characterize the Nöbeling space of dimension \(k\). The author proves that this characterization holds for other values of \(k\) as well.
Theorem. For every \(2 \leq k < \infty\), if a strongly \(k\)-universal \(k\) dimensional Polish space is an absolute extensor in dimension \(k\), then it is homeomorphic to \(N_{k}^{2k+1}\).
In his approach to the proof of this result, the author formulates an axiom system of Nöbeling spaces which concerns partition refinements and partition transformations, such that two arbitrary separable metric spaces, with the same finite dimension and connectivity properties, satisfying these axioms are homeomorphic. The author indicates that it is possible to prove the analogous characterization of the Menger spaces through a similar axiomatic approach.
The present first part of the series of three papers contains the axiom system of the Nöbeling spaces, and several surgery techniques which allow to improve the connectivity properties of partitions. In the second part, the author proves several technical results related to partitions and approximations of maps. The final third part contains the proof of the consistency of the axiom system of Nöbeling spaces, i.e. the author shows that the so-called Nöbeling cores of constructible manifolds satisfy these axioms.

Reviewer: Tamás Mátrai (Toronto)

MSC:

54F65	Topological characterizations of particular spaces
54C55	Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54F45	Dimension theory in general topology
55P15	Classification of homotopy type

Keywords:

axiom system of Nöbeling spaces; Menger space; sigma product; strongly universal \(k\) dimensional Polish space; absolute extensor; constructive sets; partition connectivity; Nöbeling core of constructive manifolds