The lattice theoretic part of topological separation properties

Abstract

In this paper we show that for each n£{2,3,4,5} the topological separation property T n can be decomposed where C, N 2 ,..., N n are purely lattice theoretic properties with the expected implications holding between them.The property C is discussed briefly in § 1 where it is explained that C is the lattice theoretic analogue of ring theoretic semisimplicity and also related to the topological property TV The four properties N 2 ,..., N 5 are discussed in §2. The properties N 4 , N 5 are the lattice theoretic analogues of topological normality and complete normality. For this reason we call N 2 ,..., N 5 normality properties. In §3 we establish the above decomposition and show that C + N n can be thought of as the lattice theoretic part of T n . In particular we show that a space has C + N n if and only if its open set lattice is isomorphic to the open set lattice of a T n space. These results extend the remarks of Davis (1, §3). Some of these results are also related to results of (2) and (3). Property C + N 3 is the lattice theoretic analogue of topological regularity, and so we call C + N 2 ,... ,C + N s regularity properties. Finally in §4 we say a few words about the T, case.

Foundations for Computable Topology

“…This is because the computable topology on the discrete space N consists of the recursively enumerable subsets, not all of them. Also, our definition of Hausdorffness is slightly different from that in locale theory [Sim78] …”

Regularity, Fitness, and the Block Structure of Frames

“…Showing the existence of such a decomposition was part of the purpose of [10]. The other was to show that in certain circumstances T 1 could be replaced by T 0 + subfit.…”

“…Property (i) appeared as axiom T 0 1 in [3]. In [10] I suggested the name Fconjunctive_ (because it is the opposite of the disjunctive property for distributive lattices), but the duller terminology seems to be more popular. Condition (i) is often used as the characterizing property.…”

“…I suspect that in each case the missing property is entirely point-free. Perhaps the analysis in [10] will help with these problems. The use of the frame 3 might seem a little odd, but there is something going on here.…”

“…The property Nð0Þ is precisely the property N 3 of [10]. The property Fð0Þ is just the standard notion of regularity.…”