Taxonomies of Model-Theoretically Defined Topological Properties

A topological classification scheme consists of two ingredients: (1) an abstract class X of topological spaces; and (2) a "taxonomy", i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class. X, is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in X is characterized within the classification scheme if whenever Y E X, and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them. ?0. Introduction. By a "taxonomy of topological properties", we mean a set {Pi: i E I} of topological properties defined and indexed in some particularly wellorganized way. What we have in mind here is that each Pi should be specified via the mechanisms of first order logic. As a prototype, let OF be the set of first order sentences in the alphabet { v, A, IT} of bounded lattices. For each o E OF, let a topological space X have property PI just in case the lattice F(X) of closed subsets of X satisfies q in the usual sense of model theory [6]. The set {IP: q E O F} is a "taxonomy" in our sense of the word, namely the closed set taxonomy TF. More formally, we define a taxonomy to be a triple T = , where: (i) P is a set of first order sentences over an alphabet L = L.i of finitary relation and operation symbols; (ii) R is a first order representation (see [2], [3], [4]), assigning to each topological space X an L-structure R(X) in such a way that R(X) and R(Y) are isomorphic structures whenever X and Y are homeomorphic spaces; and (iii) k=* is a "satisfaction relation" between L-structures and members of '. (We do not seek to axiomatize satisfaction relations here; however we would certainly Received June 10, 1988; revised April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 03C15, 03C20, 06D99, 54D30, 54F15, 54F25. ? 1990, Association for Symbolic Logic 0022-4812/90/5502-001 2/$02.50