Abstract. We develop a powerdomain construction, [.], which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains. The desire for such a construction arises when considering programming languages with nondeterministic features or parallel features treated in a nondeterministic way. We hope to achieve a natural, fully abstract semantics in which such equivalences as (pparq)=(qparp) hold. The domain (D Truthvalues) is not the right one, and instead we take the (finitely) generable subsets of D. When D is discrete they are ordered in an elementwise fashion. In the general case they are given the coarsest ordering consistent, in an appropriate sense, with the ordering given in the discrete case. We then find a restricted class of algebraic inductive partial orders which is closed under [. as well as the sum, product and exponentiation constructions. This class permits the solution of recursive domain equations, and we give some illustrative semantics using 5[.]. It remains to be seen if our powerdomain construction does give rise to fully abstract semantics, although such natural equivalences as the above do hold. The major deficiency is the lack of a convincing treatment of the fair parallel construct. 1. Introduction. When one follows the Scott-Strachey approach to the

We study modules over the ring ˜ C of complex generalized numbers from a topological point of view, introducing the notions of ˜ C-linear topology and locally convex ˜ C-linear topology. In this context particular attention is given to completeness, continuity of ˜ C-linear maps and elements of duality theory for topological ˜ C-modules. As main examples we consider various Colombeau algebras of generalized functions. Key words: modules over the ring of complex generalized numbers, algebras of generalized functions, topology, duality theory

In this paper we begin to develop the filter approach to (completeness of) quasiuniform spaces, proposed in [8, Section V]. It will be seen that this permits a more powerful and elegant account of completion to be given than was feasible using sequences or nets [8].

The Kuratowski Closure-Complement Theorem 1.1. [29] If (X, T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained.

Introduction Convergence and continuity properties of homomorphisms play an important role in the theory of topological projective planes. Grundhofer [8] showed that the set \Sigma of all automorphisms of a compact projective plane is a locally compact transformation group with respect to the topology of uniform convergence; for the special case of compact connected projective planes see also Salzmann [21]. With regards to classification, compact connected projective planes have been successfully investigated by studying their automorphism group, see Salzmann, Betten, Grundhofer, Hahl, Lowen, and Stroppel [22] for a detailed exposition. Salzmann [20] proved that if \Pi is a 2-dimensional compact projective plane, then on \Sigma the topology of pointwise convergence coincides with the topology of uniform convergence. He also showed ([19]) that any homomorphism between 2-dimensional compact projective planes is in fact a homeomorphism. Grundhofer [9] characterized the continuity of non-

this paper, we first present a survey of existing results on nonsequential systems within the framework of Witsenhausen's intrinsic model; then, we discuss some open problems arising from the research performed so far. 1. Introduction

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms [BOst11], we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We brie‡y study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowlyvarying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata [BajKar] from groups to ‡ows on a group. A multiplicative representation of the ‡ow derived in [Ost-knit] demonstrates equivalence of the ‡ow with the earlier group formulation. In 1 companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in [BOst14] and we extend to locally compact,-compact groups the fundamental theorems on characterization and representation [BOst15]. In [BOst16], working with topological ‡ows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be speci…ed using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure. Classi…cation: 26A03

We study modules over the ring ˜ C of complex generalized numbers from a topological point of view, introducing the notions of ˜ C-linear topology and locally convex ˜ C-linear topology. In this context particular attention is given to completeness, continuity of ˜ C-linear maps and elements of duality theory for topological ˜ C-modules. As main examples we consider various Colombeau algebras of generalized functions. Key words: modules over the ring of complex generalized numbers, algebras of generalized functions, topology, duality theory

We generalize the Serre-Swan theorem to non-commutative C ∗-algebras. For a Hilbert C ∗-module X over a C ∗-algebra A, we introduce a hermitian vector bundle EX associated to X. We show that there is a linear subspace ΓX of the space of all holomorphic sections of EX and a flat connection D on EX with the following properties: (i) ΓX is a Hilbert A-module with the action of A defined by D, (ii) the C ∗-inner product of ΓX is induced by the hermitian metric of EX, (iii) EX is isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv) ΓX is isomorphic to X.

We study some measures which are related to the notion of the ǫ-complexity. We prove that measure of ǫ-complexity defined on the base of the notion of ǫ-separability is equivalent to the dual measure that is defined through ǫ-nets.