A topological position of the set of continuous maps in the set of upper semicontinuous maps
Let \((X, \rho )\) be a metric space and \(\downarrow \text{USCC}(X)\) and \(\downarrow \text{CC}(X)\) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from \(X\) to \(\mathbf I = [0,1]\), respectively. Both spaces are endowed with the Hausdorff-metric. In the paper under review, the authors prove that, if \(X\) is an infinite compact metric space with a dense set of isolated points, then \((\downarrow \text{USCC}(X), \downarrow \text{CC}(X)) \approx (Q, c _{0} \cup (Q \backslash \Sigma ))\), i.e., there is a homeomorphism \(h:\downarrow \text{USCC}(X) \rightarrow Q\) such that \(h(\downarrow \text{CC}(X)) = c _{0} \cup (Q \backslash \Sigma )\), where \(Q = [ - 1,1]^{\omega }, \Sigma = \{(x _{n})_{n\in \mathbb N} \in Q: \sup|x _{n}| < 1\}\) and \(c_{0} = \{(x _{n })_{n\in \mathbb N} \in \Sigma : \lim_{n\rightarrow +\infty } x _{n } = 0\}\). Combining this statement with the main result in the paper by \textit{Z. Yang} and \textit{X. Zhou} [Topology Appl. 154, No. 8, 1737--1747 (2007; Zbl 1119.54010)], the authors obtain the following theorem, which is a very nice and complete description of the pair \((\downarrow \text{USCC}(X), \downarrow \text{CC}(X))\), for a compact metric space \(X\): \[ (\downarrow \text{USCC}(X), \downarrow \text{CC}(X)) \approx \begin{cases} (Q,c_0 \cup (Q\backslash \Sigma )), &\text{if the set of isolated points is dense in \(X\)}, \\ (Q,c_0 ), \quad &\text{otherwise},\end{cases} \] if \(X\) is an infinite compact metric space. They also prove that, for a metric space \(X, (\downarrow \text{USCC}(X), \downarrow \text{CC}(X)) \approx (\Sigma , c _{0})\) if and only if \(X\) is non-compact, locally compact, non-discrete and separable.
scientific article; zbMATH DE number 5643468
scientific article