One-point compactifications and continuity for partial frames

Locally compact Hausdorff spaces and their one-point compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results.Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.Partial frames are meet-semilattices in which not all subsets need have joins.A distinguishing feature  of their study is that a small collection  of axioms of an elementary nature allows one to do much that is  traditional for frames or locales. The axioms are sufficiently general to include as examples \sigma-frames, \kappa-frames and frames.In this paper, we present the notion of a continuous partial frame by means of a suitable ``way-below'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a one-point compactification for a regular continuous partial frame using generators and relations.  We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional. We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r\^{o}le here. The paper concludes with a proof of the uniqueness of the one-point compactification.

Keywords:

frame , partial frame , sels-frame , kappa-frame , sigma-frame , mathcal{Z}-frame , compactification , one-point compactification , strong inclusion , strongly regular ideal , continuous lattice , locally compact