Page 77 - Textos de Matemática Vol. 41
P. 77
Chapter 10
Separation I. Regular, fit and subfit
10.1. Regular, fit and subfit locales. Regarding the standard separation axioms of classical topology
T0,T1,T2,...,
the first cannot have any point-free counterpart: two points violating it can- not be told apart by open sets. With respect to T1, the classical situation is almost so heavily dependent on points that we cannot expect an exact counter- part. This chapter is concerned with fitness and subfitness, point-free properties loosely related to T1. Regularity, on the contrary, is an extension of the classical homonymous property.
A locale (frame) L is said to be regular if
ab ⇒ ∃c, a∨c=1 and c→0b.
Itissaidtobefit if
ab ⇒ ∃c, a∨c=1 and c→bb
and, finally, to be subfit if
ab ⇒ ∃c, a∨c=1̸=b∨c.
(Reg)
(Fit)
(Sfit)
Note. The regularity above is an extension of the homonymous property from classical topology. That is, a space X is regular in the classical sense iff Lc(X) is regular in the sense above (the condition can be translated to stating that for open U, V such that U V there is an open W such that W ⊆ U and W V ; see also 11.6.2 below).
63
