  We examine the analogues for the respective categories of locales of two
well-known results about regularity and effectiveness of some categories of
spaces. We show that the category of compact regular locales is effective
regular (=Barr-exact). We also show that the category of compactly generated
Hausdorff locales is regular, provided that it is coreflective within Hausdorff
locales. We do not appeal to the existence of points (which would render the
two results trivial) but rely on the treatment of the subject by methods that
are valid in the internal logic of a topos. On the course to the result about
compactly generated locales we arrive at a generalization of a result of B. Day
and R. Street, deriving regularity for a cocomplete category containing a dense
regular subcategory closed under finite limits and colimits and satisfying a
certain compatibility condition of pullbacks with appropriate colimits.

Category Theory