On the coincidence of the upper Kuratowski topology with the cocompact topology (Q1296269)

Let \(X\) be a topological space and denote by \(C(X)\) the hyperspace of \(X\), i.e. \(C(X)\) is the set of all closed subsets of \(X\). The authors consider in \(C(X)\) standard topologies and notions of convergence: the Fell topology \(\tau_F\) and the Hausdorff (or Kuratowski) convergence. Within the lattice of all topologies for \(C(X)\) we can write \(\tau_F=\tau_{uF}\vee \tau_l\), where \(\tau_{uF}\) is the upper Fell topology and \(\tau_l\) is the lower semi-finite topology, defined by \textit{E. Michael} [Trans. Am. Math. Soc. 71, 152-182 (1951; Zbl 0043.37902)]. The authors call \(\tau_{uF}\) the cocompact topology \(\tau_{co}\), \(\tau_{co}=\tau_{uF}\). The Hausdorff or Kuratowski convergence is given by the \(\lim\sup\)-convergence and the \(\liminf\)-convergence: let \((A_i)_{i\in I}\) be a net from \(C(X)\), \(A\in C(X)\); following the notions of the authors, \((A_i)\) upper Kuratowski converges to \(A\), \(A_i\overset{uK}{\longrightarrow}A\), iff \(\lim\sup A_i\subseteq A\), \((A_i)\) converges lower Kuratowski to \(A\), \(A_i\overset{lK}{\longrightarrow}A\), iff \(A\subseteq\lim\inf A_i\), \((A_i)\) converges Kuratowski to \(A\), \(A_i\overset{K}{\longrightarrow}A\), iff \(\lim\inf A_i=\lim\sup A_i=A\). \(uK\)-convergence is in general not topological, but \(lK\) is equal to the convergence of \(\tau_l\) [see the reviewer, Fundam. Math. 59, 159-169 (1966; Zbl 0139.40404)]. Hence we can write \(K=uK\vee\tau_l\). Then the topology associated to \(uK\) and to \(K\) respectively is called upper Kuratowski topology \(\tau_{uK}\) and convergence topology \(\tau_c\) respectively. It is known that \(\tau_{co}\leq \tau_{uK}\) holds, and from this fact one gets \(\tau_F\leq\tau_c\). \textit{S. Dolecki, G. H. Greco} and \textit{A. Lechicki} say that a space \(X\) is consonant if \(\tau_{co}=\tau_{uK}\) holds and that \(X\) is dissonant if not [Pac. J. Math. 117, 69-98 (1985; Zbl 0511.54012); C. R. Acad. Sci., Paris, Sér. I 312, No. 12, 923-926 (1991; Zbl 0789.54009)]. \textit{M. Arab} and \textit{J. Calbrix} say that a space is hyperconsonant if in \(C(X)\) \(\tau_F=\tau_c\) holds [ibid. 318, No. 6, 549-552 (1994; Zbl 0795.54015); Set-Valued Anal. 5, No. 1, 47-55 (1997; Zbl 0876.54007)]. The mentioned authors and some others obtained a number of results concerning these notions; to mention few examples: \textit{S. Dolecki, G. H. Greco} and \textit{A. Lechicki} proved that every Čech-complete space is consonant; \textit{T. Nogura} and \textit{D. Shakhmatov} [Topology Appl. 70, No. 2-3, 213-243 (1996; Zbl 0848.54007)] proved that every hyperconsonant space is consonant. In the present paper the authors continue the study of consonant and hyperconsonant spaces using some new tools (a criterion of consonance which is based on the property of sequentiality, the application of the concept of Radon measures). They are able to get new results, solving by this way some problems which were posed in papers of the preceding contributors to the subject. For example, the authors prove that the Sorgenfrey real line is dissonant, giving a negative answer to a question of S. Dolecki, G. H. Greco and A. Lechicki.