On some notions related to compactness for locales.

*(English)*Zbl 0698.54017A topological space is weakly locally compact if each point has a compact closed neighbourhood. The authors define, by analogy with an appropriately chosen characterization of weakly locally compact topologies, the corresponding notion for locales. These locales are shown to behave in the same way as their spatial counterparts under products, sums and the taking of closed sublocales. When almost compact, the weakly locally compact locales are compact. Having defined a one point extension \(L_ F\) of a locale L as a sublocale of \(L+2\) generated by the disjoint union of L and a filter F on L, conditions on L and F are given which determine, amongst other things, when \(L_ F\) is spatial, regular, Hausdorff or compact. As a consequence it is proved that each regular weakly locally compact locale is spatial and completely regular.

Reviewer: C.R.A.Gilmour

##### MSC:

54D45 | Local compactness, \(\sigma\)-compactness |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

54D30 | Compactness |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

06B30 | Topological lattices |