Topological structures of the space of continuous functions on a non-compact space with the Fell topology (Q2850618)

Let \(X\) be a metrizable space and \(C(X)\) the set of all continuous real-valued functions from \(X\) to the unit interval \(I.\) Let \(USC(X) = USC(X,I)\) consist of all upper semi-continuous maps from \(X\) to \(I.\) For every \(f \in USC(X)\) let NEWLINE\[NEWLINE\downarrow f = \{(x,t) \in X \times I : t \leq f(x) \} NEWLINE\]NEWLINE be the hypograph of \( f\) and, for every \( A \subset USC(X)\), NEWLINE\[NEWLINE \downarrow A = \{ \downarrow f : f \in A \} NEWLINE\]NEWLINE which is a subset of \( \;Cld(X \times I),\) the hyperspace of all non-empty closed sets of \(X \times I.\) The paper investigates the pair of function spaces \(( \downarrow USC(X), \downarrow C(X))\) with various topologies, but in particular when carrying the Fell topology, in that case denoted as \( USC_{F}(X)\) and \(C_{F}(X),\) respectively. The choice of the Fell topology is justified from its independence of any admissible metric of the underlying space \(X\). The authors perform as main issue the following one:NEWLINENEWLINEIf \(X\) is a non-compact, locally compact, separable and non-dicrete metrizable space with \( { \overline{X}}_{0} = X,\) then NEWLINE\[NEWLINE ( \downarrow USC_{F}(X), \downarrow C_{F}(X)) \thickapprox (Q, c_{0} \cup (Q \setminus \Sigma)),NEWLINE\]NEWLINE where \({\overline{X}}_{0}\) denotes the closure of the set of isolated points in \(X, \;Q = [0,1]^{\omega}\) is the Hilbert cube, \(\Sigma = \{ (x_{n}) \in Q : sup | x_{n}| < 1 \}\) and \(c_{0} = \{ (x_{n}) \in \Sigma : \lim_{n \to \infty}x_{n} = 0 \} \); and, finally, where \((X,Y) \thickapprox (A,B)\) means that there exists a homeomorphism \(h: X \rightarrow A\) such that \(h(Y) = B.\)NEWLINENEWLINEThe authors achieve the main result by means of topological semilattices and by using fine techniques of algebraic topology applied to the players in the triple: NEWLINE\[NEWLINE( \downarrow USC_{F}(X), \downarrow USC_{F}(X) \;\setminus \downarrow C_{0}(X), \downarrow C_{F}(X) \;\setminus \downarrow C_{0}(X))NEWLINE\]NEWLINE where \(C_{0}(X) = \{ \;f \in USC(X) : f(x) = 0, \;\;\forall x \in X' \}\), with \(X'\) the derived set of \(X,\) and by proving that: {\parindent=6mm \begin{itemize}\item[1)] \(\downarrow USC_{F}(X) \thickapprox Q,\) \item[2)] \(\downarrow USC_{F}(X) \;\setminus \downarrow C_{0}(X) = \bigcup _{n=1}^{\infty} \downarrow F_{n}(X)\) is the union of a tower \((\downarrow F_{n}(X))_{n}\) in \(\downarrow USC_{F}(X)\) such that: {\parindent=12mm \begin{itemize}\item[a)] \(\downarrow F_{n}(X) \in Z(\downarrow USC_{F}(X)) \cap Z( \downarrow F_{n+1}(X)),\) \item[b)] \(( \downarrow F_{n}(X), \downarrow F_{n}(X) \cap (\downarrow C(X) \;\setminus C_{0}(X))) = (\downarrow F_{n}(X), \downarrow C_{n}(X))\) is strongly \(({\mathcal M}_{0}, {\mathcal F}_{\sigma\delta)}-\) universal for every \(n \in \omega\) and \item[c)] \(\downarrow USC_{F}(X) \;\setminus \downarrow C_{0}(X)\) is a capset for \(\downarrow USC_{F}(X)\). NEWLINENEWLINE\end{itemize}}NEWLINENEWLINE\end{itemize}} So, consequently, getting: NEWLINE\[NEWLINE (\downarrow USC_{F}(X), \downarrow USC_{F}(X) \;\setminus \downarrow C_{0}(X), \downarrow C_{F}(X) \;\setminus \downarrow C_{0}(X)) \;\;\thickapprox (Q, \Sigma,c_{0}).NEWLINE\]