The Hofmann-Mislove Theorem for general topological structures

In this paper we prove a modification of Hofmann-Mislove theorem for a topological structure similar to the minusspaces of de Groot, in which the empty set "need not be open". This will extend, in a slightly relaxed form, the validity of the classical Hofmann-Mislove theorem also to some of those spaces, whose underlying topology need not be (quasi-) sober

How regular can maxitive measures be?

We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every regular maxitive measure is completely maxitive, this yields sufficient conditions for the existence of a cardinal density. We also show that every outer-continuous maxitive measure can be decomposed as the supremum of a regular maxitive measure and a maxitive measure that vanishes on compact subsets under appropriate conditions.Comment: 24 page

Probabilistic Monads, Domains and Classical Information

Shannon's classical information theory uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results of Martin, Allwein and Moskowitz for binary channels to show how some more modern tools - probabilistic monads and domain theory in particular - can be used to model classical channels. As initiated Martin, et al., the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the (n x n)-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.Comment: In Proceedings DCM 2011, arXiv:1207.682

04351 Abstracts Collection -- Spatial Representation: Discrete vs. Continuous Computational Models

From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 ``Spatial Representation: Discrete vs. Continuous Computational Models\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Topological Scott Convergence Theorem

Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of $T_0$ spaces instead of restricting to posets. In this paper, we respond to this calling with an attempt to formulate a topological version of the Scott Convergence Theorem, i.e., an order-theoretic characterisation of those posets for which the Scott-convergence $\mathcal{S}$ is topological. To do this, we make use of the $\mathcal{ID}$ replacement principle to create topological analogues of well-known domain-theoretic concepts, e.g., $\mathcal{I}$-continuous spaces correspond to continuous posets, as $\mathcal{I}$-convergence corresponds to $\mathcal{S}$-convergence. In this paper, we consider two novel topological concepts, namely, the $\mathcal{I}$-stable spaces and the $\mathcal{DI}$ spaces, and as a result we obtain some necessary (respectively, sufficient) conditions under which the convergence structure $\mathcal{I}$ is topological

On Maximality of Compact Topologies

Using some advanced properties of the de Groot dual and some generalization of the Hofmann-Mislove theorem, we solve in the positive the question of D. E. Cameron: Is every compact topology contained in some maximal compact topology