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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.

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a b s t r a c t I describe a simple construction that assigns a pre-nucleus and associated nucleus to each frame. The nature of this nucleus appears to measure the subfitness properties of the parent frame. I pose a few questions concerning this pair.