# Freeness and continuity in semilattices

It is generally known that the set of finite subsets F(X) of a set X forms a lattice under the operations of union and intersection, and that F(X) with union as the principal operation is the free semilattice on the set X. Michael, in {43} topologized (GAMMA)(X), the space of closed subsets of a topological space X, with an intrinsic topology arising from the topology on X itself and obtained results of a topological nature about (GAMMA)(X) and F(X). We denote this topology, called the finite topology by Michael and the exponential topology by Kuratowski {35}, by v in honor of L. Vietoris {57} Lawson in {37} studied topological semilattices which have a basis of subsemilattice neighborhoods at each point. We call these Lawson semilattices. We show in Chapter I that (F(X), v) is the free Lawson semilattice on the space X. We give F(X) a second, finer topology, denoted by t, which makes F(X) into the free topological semilattice on X. From the relationship between these topologies we develop the study of the topology t. These results parallel some of those of B. V. S. Thomas in her study {55} of free topological groups and exemplify the statement by O. Wyler {60} that any free algebra functor lifts to a free topological algebra functor We introduce a third topology k on F(X) which refines t and make F(X) into the free k-semilattice on the k-space X. The relation v (L-HOOK) t (L-HOOK) k leads to results about the free k-semilattice which are similar to the findings of W. F. LaMartin in his study {36} of free k-groups Continuous lattices were defined by Scott in {47} in order to find a model for the Church-Curry (lamda)-calculus in logic and subsequently to lay the foundation for a theory of computation in {48}. His concept was shown to be equivalent to the already existing idea of a compact Lawson semilattice in topological algebra after the work of Hofmann and Stralka {27} In Chapter II we study the closure operators on a continuous lattice and show that the construction of the closure operator space is functorial. Moreover, the closure operators determine the subalgebras of a continuous lattice. Scott's work {47} showed that a function space functors on his category of continuous lattices preserved projective limits. His construction was amplified and elucidated in {19}. We show that the closure operator functor is a subfunctor of Scott's functor and also preserves projective limits. Using this fact, we show that each continuous lattice is the quotient of a continuous lattice which is isomorphic to its space of closure operators Complete Heyting algebras were shown to be important in intuitionistic logic by Skolem in {52}, in sheaf theory by Fourman and Scott in {15}, in topos theory by Freyd in {16}, and in an investigation of the algebraic nature of topology (under the name frames) by Dowker and Strauss in {7}, {8}, and {9}. D. S. Macnab studied the algebraic theory of modal operators on a complete Heyting algebra in {41} and {42} We examine the continuous Heyting algebras in Chapter III and combine the study of continuous lattices with that of complete Heyting algebras. We show that the Scott continuous modal operators on continuous Heyting algebras determine the subalgebras just as the closure operators do for continuous lattices. Furthermore, the construction of the modal operator space is functorial. These results parallel those of Chapter II. One open question which remains is whether the Scott continuous modal operator space on a continuous Heyting algebra is again a continuous Heyting algebra. We offer an affirmative answer in the case that the <<-relation on the algebra is multiplicative