On the representations of an abstract lattice as the family of closed sets of a topological space
David A. Drake, W. J. Thron · Transactions of the American Mathematical Society · 1965
Introduction. In 1962 Thron considered certain questions concerning the lattice structure of the family # of closed sets of a topological space.In this article we continue these investigations.Before describing the results of this paper it will be helpful to review certain definitions and give a few new ones.Whenever we talk about a topological space (X, *£) we shall mean by <& the family of closed sets on X.Clearly, it defines the topological structures as well as the family of open sets, and for our purposes it is more convenient.Let (Ty, 2: x) and (r2, ^2) be two lattices; then a function / from Tx onto T2 will be called an isomorphism iff / is 1-1 and / as well as its inverse are order preserving.An isomorphism from T onto itself will be called an automorphism on T. It is well known that/ is an isomorphism iff it and its inverse preserve l.u.b. and g.l.b., that is /( V[«.])= V[/(«.)] and /(AM) = A[M)] and similarly for/-1.A subset A of a lattice (r, ïï) is called a base of T iff every element a e T, other than the least element if it exists in T, can be written as a = \J[di:d¡eA,ieIa].It is convenient for our purposes to assume that if T has a least element then this element does not belong to any base of T. Observe that we are not asserting that, for every subcollection ficA, \/[d: deÙ] exists in T.An element a of a lattice (T, ^) will be called irreducible (strongly irreducible) iff a cannot be expressed as the l.u.b. of a finite (arbitrary) number of elements of T, which are strictly less than a.A lattice is called a set lattice iff its elements are sets and the order relation is given by set inclusion.We note that, whenever V[^-¡] and A [CJ exist then V[cj = |J[cJand AMsflM-A set lattice in which for finite index sets /, V [Q : i e /] = (J [C¡ : i e /] and A [C¡: i e 7] = 0 [C¡ : i e f] is called a proper set lattice.A set representation of a lattice (r, ^) is an ordered pair iff!, 2),/), where (f€, 3) is a set lattice and/