The topology of systems of hyperspaces determined by dimension functions (Q968875)

Let \(\Gamma\) be an index set. \textit{A \(\Gamma\)-system} \(\langle X, X_\gamma \rangle_{\gamma \in \Gamma}\) is a pair consisting of a space \(X\) and a family \(\langle X_\gamma \rangle_{\gamma \in \Gamma}\) of its subsets. The \(\Gamma\)-systems \(\langle X, X_\gamma \rangle_{\gamma \in \Gamma}\) and \(\langle Y, Y_\gamma \rangle_{\gamma \in \Gamma}\) are homeomorphic if there is a homeomorphism \(h: X \to Y\) such that \(h(X_\gamma) = Y_\gamma\) for each \(\gamma \in \Gamma.\) By \textit{a dimension function} the authors mean a function \(D: 2^X_* \to [0, \infty]\) defined on the family \(2^X_*\) of compact subsets of a topological space \(X\) to the half-line \([0, \infty]\) with infinity attached such that (1) \(D(\emptyset) = 0\); (2) \(D(A) \leq D(B)\) whenever \(A \subset B\); (3) \(D(F \cup A \cup B) \leq \max\{D(A), D(B)\}\) whenever the set \(F\) is finite and the sets \(A, B\) are disjoint; (4) every non-empty open set \(U \subset X\) contains non-empty open sets \(U_n \subset U, n \in \omega,\) such that each compact set \(K \subset \text{Cl}_X(\bigcup_{n \in \omega} U_n)\) has \(D(K) \leq \sup_{n \in \omega} D(K \cap \overline{U}_n).\) Let \(\Gamma \subset [0, \infty]\). For each \(\gamma \in \Gamma\) put \(D_{\leq \gamma} (X) = \{F \in 2^X : D(F) \leq \gamma\}.\) Let further \(Q\) be the Hilbert cube \([-1,1]^\omega\), \(s = (-1,1)^\omega\) its pseudointerior, \(\mathbb Q\) the space of rational numbers and \(\mathbb Q _{\leq \gamma} = \{q \in \mathbb Q : q \leq \gamma\}\), \(\mathbb Q _{> \gamma} = \{q \in \mathbb Q : q > \gamma\}\) for each real number \(\gamma\). The main theorem of the paper gives necessary and sufficient conditions on the space \(X\), providing that the \(\Gamma\)-system \(\langle 2^X, D_{\leq \gamma} (X) \rangle_{\gamma \in \Gamma}\) is homeomorphic to the \(\Gamma\)-system \(\langle Q^\mathbb Q, Q^{\mathbb Q _{\leq \gamma}} \times s^{\mathbb Q _{> \gamma}} \rangle_{\gamma \in \Gamma}\). The theorem can be applied to such functions as the covering dimension \(\dim\), the cohomological dimension \(\dim_G\), where \(G\) is an arbitrary Abelian group, and the Hausdorff dimension \(\dim_H\). In particular, the characterization of the spaces \(X\) for which the system \(\langle 2^X, \dim_{\leq n} (X) \rangle_{n \in \omega}\) is homeomorphic to the system \(\langle Q^\omega, Q^n \times s^{\omega \setminus n} \rangle_{n \in \omega}\) from \textit{R. Cauty} [Fundam. Math. 159, No.~2, 115--126 (1999; Zbl 0930.57024)] is one of the corollaries of the main theorem. Another one is the following Mean Value Theorem for the Hausdorff dimension \(\dim_H\) saying that for every separable complete metric space \(X\) and every non-negative real number \(d < \dim_H(X)\) the space \(X\) contains a compact subset \(K \subset X\) with \(\dim_H(K) = d\).