On d-Spaces

The article is devoted to the investigations of the topological properties of d-spaces introduced by Wyler, Lecture Notes in Mathematics, vol. 871 (1981), 384-389. In particular, it is established that for this rather wide class of topological spaces the known Scott construction gives the spaces that are homeomorphic to the spaces of all continuous mappings into themselves. The present article is devoted to mathematical foundations of denotational semantics. It is shown that d-spaces introduced in [l] (in [2], they are called the monotone convergence spaces) have natural closedness properties (Propositions 2-4) and behave well with respect to inverse limits of inverse spectra of projections (Propositions 6 and 7). Thus, one can apply the well-known Scott construction [3] of topological spaces X that are homeomorphic to the space C(X,X) of all continuous mappings of X into itself. The class of d-spaces is sufficiently wide. It contains all Tl-spaces and all sober spaces [2]. Proposition 5 shows that for any inverse spectrum of projections the " dual " direct spectrum has direct limit which is homeomorphic to a subspace of the inverse limit, being its d-basis. Theorem 1 yields a necessary (and sufficient) condition for the " continuity " of a functor on the inverse limits in the category 9'p of d-spaces with projections taken as morphisms. As a conclusion, the following fact is established: For any To-space X and any its extension Y, being a d-space, the smallest d-space X0 C Y containing X is uniquely determined up to a homeomorphism on X. As a consequence, we obtain Wyler's result [l, Theorem 2.71 about the existence of d-completions. Hereinafter, all topological spaces are assumed to be separated (To-spaces). Then the preorder <x on the topological space X (the subscript X will be often omitted) defined by the topology as follows: for <,<' E X 4 <X t' % for any open U C X(5 E U =+ <' E U) (cf., for example, [2]) is a (partial) order.