Meets of spatial sublocales (Q983856)

As \textit{P.\,T.\,Johnstone} stressed in [``The art of pointless thinking: a student's guide to the category of locales'', Res. Expo. Math. 18, 85--107 (1991; Zbl 0745.18003)], ``one of the most important reasons for studying the category of locales as a substitute for (and in many ways an improvement on) the category of topological spaces'' (which was indeed the reason that originally moved \textit{J.\,R.\,Isbell} in [``Atomless parts of spaces'', Math. Scand. 31, 5--32 (1972; Zbl 0246.54028)] to study locales) is the fact that ``locale products are so much nicer than space products: all sorts of topological properties which fail to be inherited by product spaces behave well for product locales''. This discrepancy lies in the difference in intersections: the canonical functor \(\Omega: \mathbf{Top} \to \mathbf{Loc}\) from the category of topological spaces and continuous maps to the category of locales and localic maps does not preserve meets of subspaces. Moreover, the meet of a family of spatial sublocales of a locale is not in general spatial. The main aim of the article under review is to characterize those topological spaces for which \(\Omega\) preserves meets of subspaces. As a byproduct, some characterizations for the meet of certain spatial sublocales of a locale to be spatial are obtained. Reviewer's remark: There is a mistake in the last sentence of Example 2.2; in fact, the space \((\mathbb{R},\tau)\) does not satisfy the conditions of Corollary 2.1 since not all open sets in \(\tau\) are closed.