Reflexive nests of finite subsets of a Banach space (Q401088)

Given a topological space \(X\), let \({\mathcal C}(X)\) be the set of all continuous endomorphisms on \(X\) and \({\mathcal S}(X)\) the set of all closed subsets of \(X\). To a subset \({\mathcal L}\) of \({\mathcal S}(X)\) we associate the set of functions \(\text{alg}{\mathcal L}=\{f\in {\mathcal C}(X)\,:\,f(A)\subseteq A\,\text{ for all }A\in {\mathcal L}\}\). Similarly, for a subset \({\mathcal F}\subseteq{\mathcal C}(X)\), define \(\text{lat}{\mathcal F}=\{A\in {\mathcal S}(X)\,:\,f(A)\subseteq A\,\text{ for all }f\in {\mathcal F}\}\). Borrowing concepts from the theory of linear operators, we say that \({\mathcal L}\subseteq{\mathcal S}(X)\) is reflexive if \(\text{lat}(\text{alg}{\mathcal L})={\mathcal L}\), and that \({\mathcal F}\subseteq{\mathcal C}(X)\) is reflexive if \(\text{alg}(\text{lat}{\mathcal F})={\mathcal F}\). The main result (Theorem 12) of the present paper is that nests of finite subsets of a separable, real Banach space are reflexive.NEWLINENEWLINEThe build-up to the main result focuses on the separable, real Banach space \({\mathbb R}\), the set of real numbers with the usual Euclidean metric topology. Take \(\chi=\{x_1,\,x_2,\,\ldots\}\) to be a countable dense subset of \({\mathbb R}\), and, for each natural number \(k\), set \(A_k=\{x_j\,:\,j\leq k\}\). Thus, \(A_1\subset A_2\subset A_3\subset \ldots\,\), and the collection \({\mathcal A}=\{\emptyset,\,A_1,\,A_2,\,\ldots,\,{\mathbb R}\}\) is a complete nest of closed subsets of \({\mathbb R}\). Two technical lemmas lead to Corollaries 6 and 7, which show that, for every natural number \(m\) and every number \(\alpha\in{\mathbb R}\backslash\chi\), there are functions \(f\) and \(g\) in \(\text{alg}{\mathcal A}\) such that \(f(x_{m+1})=g(\alpha)=x_m\). These results lead to the conclusion that \({\mathcal A}\) is reflexive (Theorem 10). In fact, the authors point out (Corollary 11) that the same is true of the complete nest generated by any increasing sequence of finite subsets of \({\mathbb R}\) whose union is dense in \({\mathbb R}\).NEWLINENEWLINETheorem 12 then shows that the results for nests of subsets of \({\mathbb R}\) extend to complete nests of finite subsets of any separable, real Banach space.NEWLINENEWLINESection 4 of the paper looks at the related concept of the reflexivity index of any reflexive closed set lattice \({\mathcal L}\) of subsets of a topological space \(X\), defined as \(\kappa({\mathcal L})=\min\{|{\mathcal F}|\,:\, \text{lat}{\mathcal F}={\mathcal L}\}\). In the specific case of the nests \({\mathcal A}=\{\emptyset,\,A_1,\,A_2,\,\ldots,\,{\mathbb R}\}\) from Theorem 10, the principal concrete results presented herein are that \(\kappa({\mathcal A})>1\) (Proposition 15), and that, if \(|x_{m+1}-x_m|\to0\), then \(\kappa({\mathcal A})=\infty\) (Theorem 18). An open problem is whether there is an ordering of the rationals \({\mathbb Q}\) for which the corresponding nest has a finite reflexive index.