More on Subfitness and Fitness

Abstract

The concepts of fitness and subfitness (as defined in Isbell [9]) are useful separation properties in point-free topology. The categorical behaviour of subfitness is bad and fitness is the closest modification that behaves well. The separation power of the two, however, differs very substantially and subfitness is transparent and turns out to be useful in its own right. Sort of supplementing the article [20] we present several facts on these concepts and their relation. First the "supportive" role subfitness plays when added to other properties is emphasized. In particular we prove that the numerous Dowker-Strauss type Hausdorff axioms become one for subfit frames. The aspects of fitness as a hereditary subfitness are analyzed, and a simple proof of coreflectivity of fitness is presented. Further, another property, prefitness, is shown to also produce fitness by heredity, in this case in a way usable for classical spaces, which results in a transparent characteristics of fit spaces. Finally, the properties are proved to be independent.

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

“…In particular, Johnstone and Sun Shu-Hao require that (T ′ 2 ) if 1 = a b then there are u, v such that u a, v b and u ∧ v = 0. Under subfitness they are all equivalent (see [12,Proposition 4]). Let us call the resulting conjunction naturally Hausdorff (note that this conjunction is conservative -see [6] -, that is, a space (X, τ ) is Hausdorff iff τ is naturally Hausdorff).…”

New Aspects of Subfitness in Frames and Spaces

“…Later, subfitness appeared sporadically in literature (with both a topological and logical motivation in [16], as a necessary and sufficient condition for admitting a generalized nearness in [7] -coauthored by Horst Herrlich -, and in a few other papers). In [17], it was indicated that the condition is by no means uninteresting, and quite recently ( [14]) the authors of the present article analyzed (a.o.) the role of subfitness as a supportive property in combination with other conditions, and its relation with the fitness.…”

“…Weak subfitness is weaker than subfitness and this is still weaker than T 1 in spaces. On the other hand, prefitness is already close to regularity: a frame L is prefit iff for each a ∈ L, a ≤ ( ∨ {x | x ≺ a}) * * (hence, it is "regular up to density"), see [14]. Somewhat surprisingly, prefitness does not imply subfitness.…”