Extenders and \(\varkappa\)-metrizable compacta (Q650426)

Let \(RC(X)\) denote the family of all regular closed subsets of \(X\). A \textit{\(\kappa\)-metric} on \(X\) is a function \(\rho: X\times RC(X)\to [0,\infty)\) such that (1) \(x\in F\) if and only if \(\rho(x,F)=0\) (2) \(F\subseteq G\) implies \( \rho(x,F)\geq \rho (x,G)\) for every \(F,G\in RC(X)\), (3) The function \(x\longmapsto \rho(x,F)\) is continuous, for each fixed \(F\in RC(X)\), (4) If \((F_\xi)_{\xi<\lambda}\) is a chain of regular closed sets, then \[ \rho(x, \overline{\bigcup_{\xi<\lambda} F_\xi})=\inf\{\rho(x,F_\xi):\xi<\lambda\} \] The notion of \(\kappa\)-metrizable space was introduced by \textit{E. V. Shchepin} [Russ. Math. Surv. 31, No. 5, 155--191 (1976; Zbl 0356.54026)]. \(X\) is called \textit{\(\kappa\)-metrizable} if there is a \(\kappa\)-metric on \(X\). (In the Russian literature, regular closed sets are called canonically closed. \(\kappa\) comes from the Russian word ``canonically'' in Cyrillic notion.) The main result of the paper under review is that for a compact space \(X\) the following conditions are equivalent: (1) \(\kappa\)-metrizable (2) for any embedding of \(X\) in a compact space \(Y\), there exists a continuous map \(r:Y\to S(X)\) such that \(r(x)=\delta_x\) for all \(x\in X\), (3) for any embedding of \(X\) in a compact space \(Y\), there exists an upper semicontinuous compact-valued map \(r:X\to\lambda X\) such that \(r(x)=\eta_x\) for all \(x\in X\). We say that a family \(\mathfrak{F}\) of closed sets is \textit{linked} if any two elements of \(\mathfrak{F}\) intersect. The superextension \(\lambda X\) of \(X\) consists of all maximal linked systems of closed sets in \(X\). Let \(\eta_x\) be a maximal linked system of all closed sets containing \(x\) in \(X\). The space \(S(X)\) can be represented as a subset of the product space \(\mathbb{R}^{C(X)}\) by identifying each \(\phi\in S(X)\) with \((\phi(f))_{f\in C(X)} \in\mathbb{R}^{C(X)}\). For any map \(f:X\to Y\) between compacta, the formula \[ (S(f)(\nu))(h)=\nu(h\circ f)\;\text{ where } \;\nu\in S(X),\; h\in C(X), \] defines a map \(S(f):S(X)\to S(Y)\), moreover \(S(g\circ f)=S(g)\circ S(h)\). The author shows that the covariant functor \(S\) in the category of compact Hausdorff spaces and continuous maps is weakly normal but not normal (in the sense of \textit{E. V. Shchepin} [Usp. Mat. Nauk 36, No. 3(219), 3--62 (1981; Zbl 0463.54009)]). Another result is that, a surjective map \(f:X\to Y\) is open if and only if so is the map \(S(f):S(X)\to S(Y)\).