Department of Mathematics, Shaan Xi Normal University, Xi'an 710062, People's Republic of China ; Department of Science and Engineering, Ritsumeikan University, Noji Higashi 1-1-1, Kusatsu-shi, Shiga 525, Japan
Abstract:
In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.