New Aspects of Subfitness in Frames and Spaces

Abstract

This paper contains some new facts about subfitness and weak subfitness. In the case of spaces, subfitness is compared with the axiom of symmetry, and certain seeming discrepancies are explained. Further, Isbell's spatiality theorem in fact concerns a stronger form of spatiality (T1-spatiality) which is compared with the TD-spatiality. Then, a frame is shown to be subfit iff it contains no non-trivial replete sublocale, and the relation of repleteness and subfitness is also discussed in spaces. Another necessary and sufficient condition for subfitness presented is the validity of the meet formula for the Heyting operation, which was so far known only under much stronger conditions.

Some aspects of (non) functoriality of natural discrete covers of locales

“…Then every frame homomorphism h : L 1 → L 2 is Ω(f ) for a continuous map. By the observation in 3.3, hence, every h : L 1 → L 2 lifts to a Boolean h. By Isbell's spatiality theorem in [7] (see also [13]) this also yields the following Observation. For compact sober subfit frames L 1 and L 2 , every frame homomorphism h : L 1 → L 2 lifts.…”

Remainders in pointfree topology

“…Remark 6.3. In localic terms, this means that κrLs is a codense sublocale of opm L q ( [15], called replete in [30]), that is, cpaq X κrLs ‰ 0 for every a ‰ 1 in opm L q. A crucial example in this context is the pointfree Stone-Čech compactification, introduced by Banaschewski and Mulvey in [8].…”

“…For this, recall that a space X is T D if for each x P X there is an open U Q x such that U txu is still open (clearly, T D is strictly stronger than T 0 and strictly weaker than T 1 ). A frame L is T D -spatial ( [9,30]) if L -OX for some T D -space X. In the following, CPpLq will be the set of all completely prime elements of L. By Therefore f rL Ss " Ž tf rbppqs | p P CPpLq X pL Squ " Ž tbpf ppqq | p P ΣL, p R Su.…”