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

*(English. Russian original)*Zbl 1147.54019
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 \(0 \leq k \leq \infty\),

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.

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.

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 |