Notes on direct limits of complete lattices and frames

In traditional topology, it is known that an inverse limit of continua (compact Hausdorff connected spaces) is a continuum. This paper gives corresponding results in locales: we show that a direct limit of compact normal (or, of compact normal connected) frames is normal (resp. normal connected). Some of the preparatory results hold in broader categories, and in fact we will give sufficient conditions for an inverse limit of complete lattices to be compact (or compact connected).