Core compactness and diagonality in spaces of open sets (Q2881006)

For a convergence space \(X\), let \({\mathcal O}_X\) denote the collection of all open subsets of \(X\), \([X, {\$} ] \) the Scott convergence of the complete lattice \(({\mathcal O}_X, \subseteq)\), that is, the convergence on \({\mathcal O}_X\) such that for every \(U \in {\mathcal O}_X\) and every filter \({\mathcal F}\) on \({\mathcal O}_X\), \(U \in \lim_{[X , \$]} {\mathcal F}\) if and only if every filter on \({\mathcal O}_X\) converging to a point of \(U\) contains an element of \(\{\bigcap _{O \in F} O: F \in {\mathcal F}\}\), and \(T[X, \$]\) the topological modification of \([X ,\$]\). In the paper under review, the authors investigate when \(T[X, \$]\) is core compact and characterize diagonality of the convergence space \(({\mathcal O}_X, [X, \$])\).NEWLINENEWLINEA topological space \(X\) is said to be core compact if for every \(x \in X\) and its neighborhood \(O\) in \(X\), there exists a neighborhood \(U\) of \(x\) such that every open cover of \(O\) has a finite subfamily that covers \(U\). A topological space \(X\) is said to be consonant if every Scott open set \({\mathcal A}\) in \({\mathcal O}_X\) is compactly generated, that is, there is a family \((K_i)_{i \in I}\) of compact subsets of \(X\) such that for every \(A \in {\mathcal A}\) there is \(i \in I\) such that \(K_i \subseteq A\). A space \(X\) is said to be infraconsonant if for every Scott open set \({\mathcal A}\) in \({\mathcal O}_X\), there exists a Scott open set \({\mathcal C}\) such that \(\{ C \cap D: C , D \in {\mathcal C}\} \subseteq {\mathcal A}\). Concerning the core-compactness of \(T[X, \$]\), the authors prove the following theorems: If \(X\) is topological and \(T[X, \$]\) is core compact, then \(X\) is infraconsonant; if \(X\) is a consonant topological space such that \(T[X , \$]\) is core compact, then \(X\) is locally compact; and a space \(X\) is infraconsonant if and only if the Scott topology on \({\mathcal O}_X \times {\mathcal O}_X\) is coincides with the product topology of the Scott topologies of \({\mathcal O}_X\) at \((X,X)\).NEWLINENEWLINELet \(\mathbb{F}X\) denote the set of all filters of \(X\). A convergence space \(X\) is said to be diagonal if for every map \({\mathcal S} : X \to \mathbb{F}X\) with \(x \in \lim S[x]\) for all \(x \in X\), the filter \(\bigcup _{F \in {\mathcal F}}\bigcap _{x \in F} {\mathcal S}[x]\) converges to \(x\) whenever \(x \in X\) and \({\mathcal F}\) is a filter satisfying \(x \in \lim {\mathcal F}\). The authors give a necessary and sufficient condition for a topological space \(X\) in order that the convergence space \(({\mathcal O}_X, [X, \$])\) is diagonal in terms of a variant of core-compactness. It is known that a convergence space is topological if and only if it is pretopological and diagonal. The authors prove that if a convergence space \(X\) is topological and \([X , \$]\) is pretopological, then \([X, \$]\) is topological; and that if \(X\) is a non locally compact Hausdorff space of character not exceeding the cardinality of the continuum, then \([X, \$]\) is not diagonal. They also give an example of a Hausdorff space \(X\) such that \([X,\$]\) is diagonal but not pretopological under the assumption that there exists an uncountable strongly inaccessible cardinal.