Compactification of Frames

Abstract

In a sense, this paper is a sequel to BANASCHEWSKI [l], dealing with aspects of compactification and frames that have come to the fore during the 21 years since [l] was written. A t that time, the concern was to describe the construction of compact &us-DORFF extension spaces for a given space in terms of frames, specifically as filter spaces in frames. The object of this was to explicate the lattice theoretic essence of certain familiar constructions in topology, much in the spirit of other work on frames a t the time, such as DOWKER-PAPERT [6] which I had heard presented a t the 1966 Prague Topological Symposium. In the years since then, frames have acquired an important new feature, as the constructive aspect of topological spaces (JOHNSTONE 191). I n this spirit, compactification of frames as such becomes a topic of interest, as opposed t o merely using a frame setting for describing compactification of spaces as in [ 11. Work in this latter vein is not new: thus, BAXASCHEWSKI-MULVEY [3] establieh the STONE-CECH compactification of frames, and JOHNSTONE [8] presents an alternative construction of this. Thc aim of the present note, then, is to provide a comprehensive view of all compactifications of a given frame. Considering some aspects of the STONE-CECII compactification of frames, it does not come as a surprise that this can be achieved with a tool which originated in the study of compactification of spaces, going back to FREUDE~TTIIAL [ 71. This paper was written during sabbatical leave spent a t the University of Cape Town. Financial assistcncc from that institution as well as from the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged. JOHNSTONE [ 91.We begin by recalling some background. For general information on frames, see A frame is a complete lattice L in whichfor any a E L and S & L, A and V signifying meet and join in L, as usual. A frame h m w p h i s m is a map h: L -+ i l l between frames preserving finite meets and arbi-

On Strong Inclusions and Asymmetric Proximities in Frames

“…It turns out that the resulting category of proximity frames is concretely isomorphic to that of totally bounded uniform frames (see [7]) and that the compactifiable frames are exactly those that admit strong inclusions (see [1]). In classical topology it was found useful to generalize proximities by dropping symmetry.…”

Conditions under which the least compactification of a regular continuous frame is perfect

“…Banaschewski [3] showed that if a frame L is regular and continuous, then it has a smallest strong inclusion ◭ on L given by a ◭ b ⇔ a ≺ b and either ↑a * or ↑b is compact, where ↑x = {t ∈ L : t x}, and a * is the pseudocomplement of a, i.e. the largest element of L whose meet with a is 0.…”

“…The set of all strong inclusions on a frame is in a one-to-one correspondence with the set of all compactifications of L. By a compactification of L we mean a compact regular frame M together with a dense onto map h : M → L, where a dense map h is one satisfying the condition that h(x) = 0 =⇒ x = 0. A map h is said to be codense if h(x) = e =⇒ x = e. The correspondence between strong inclusions and compactifications is described by Banaschewski [3] as follows: If ⊳ is a strong inclusion on L, we consider γL = {J : J is a strongly regular ideal of L} where an ideal J is said to be strongly regular if x ∈ J implies y ∈ J for some x ⊳ y.…”

“…the largest element of L whose meet with a is 0. Recall that a strong inclusion ⊳ on L (Banaschewski [3]) is a binary relation on L being the frame counterpart of the well known Efremovič proximity relation for spaces. The set of all strong inclusions on a frame is in a one-to-one correspondence with the set of all compactifications of L. By a compactification of L we mean a compact regular frame M together with a dense onto map h : M → L, where a dense map h is one satisfying the condition that h(x) = 0 =⇒ x = 0.…”

“…In [3] Banaschewski showed that a frame has a smallest compactification if and only if it is regular continuous. It is natural therefore to speak of regular continuous frames as the frame analogue of locally compact Hausdorff spaces, and the smallest compactification of a frame as the analogue of the Alexandroff one-point compactification.…”