Characterizations are obtained of those lattices that are isomorphic to the lattice of zero-sets of the following types of Tychonoff spaces: (i) compact (Hausdorff), (ii) Lindelöf, (iii) realcompact and normal, (iv) realcompact.

A semigroup over a generalized tree, denoted by the term ℳL-semigroup, is a compact semigroup S such that Green's relation H is a congruence on S and S/H is an abelian generalized tree with idempotent endpoints and E(S/H) a Lawson semilattice. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the generalized tree S/H in a manner similar to the construction of semigroups over trees and of the hormos. Indeed, semigroups over trees are shown to be particular examples of the construction given herein.

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

I discuss various necessary and sufficient conditions for a K-analytic space to be Souslin. In particular, I show that if the continuum hypothesis is true, then there is a non-Souslin K-analytic space in which every compact set is metrizable; while if Martin's Axiom is true and the continuum hypothesis is false, this is impossible.

Novikov [7] or [5, Th. 2, p. 510] proves that: if{An} is a sequence of analytic sets in a complete separable metric space and

then there is a sequence {Bn} of Borel sets with

§1. Introduction and Summary. Throughout X is a complete separable metric space. We write K1 for the family of non-empty compact subsets of X. K1 may be endowed with a metric (first introduced by Hausdorff) under which K1 is complete and separable. We shall make use of the subbase for this metrizable topology of K1 given by sets of the two forms

for U open in X (see Kuratowski [4] or E. Michael [9] for a discussion of topologies on the space of subsets of X). if we shall be concerned with sets in [0, 1] × X which are universal for ℋ. To define these let us make the convention that, for D ⊆ [0, 1] × X, we write

D is said to be universal for ℋ if