First steps in descriptive theory of locales

Abstract:

F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is ${F_\sigma }$ and ${G_\delta }$. This extends to arbitrary metrizable locales when "${F_\sigma }$" and "${G_\delta }$" are taken in the localic sense (countable join of closed, resp. meet of open, sublocales). In any locale, the developable sublocales are exactly the complemented elements of the lattice of sublocales. The main further results of this paper concern the strictly pointless relative theory, which exists because—always in metrizable locales— there exist nonzero pointless-absolute ${G_\delta }’{\text {s}}$, ${G_\delta }$ in every pointless extension. For instance, the pointless part ${\text {pl}}({\mathbf {R}})$ of the real line is characterized as the only nonzero zero-dimensional separable metrizable pointless-absolute ${G_\delta }$. There is no nonzero pointless-absolute ${F_\sigma }$. The pointless part of any metrizable space is, if not zero, second category, i.e. not a countable join of nowhere dense sublocales.