Separating minimal valuations, point-continuous valuations, and continuous valuations

We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations: (1) Let ${\mathcal J}$ be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on ${\mathcal J}$ with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for $U=\emptyset$ . Then, μ is a point-continuous valuation on ${\mathcal J}$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb{R}_\ell$ . Its restriction to the open subsets of $\mathbb{R}_\ell$ is a continuous valuation λ. Then, its image valuation $\overline\lambda$ through the embedding of $\mathbb{R}_\ell$ into its Smyth powerdomain $\mathcal{Q}\mathbb{R}_\ell$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $\overline\lambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.

Infinite games and quasi-uniform box products

<p>We introduce new infinite games, played in a quasi-uniform space, that generalise the proximal game to the framework of quasi-uniform spaces. We then introduce bi-proximal spaces, a concept that generalises proximal spaces to the quasi-uniform setting. We show that every bi-proximal space is a W-space and as consequence of this, the bi-proximal property is preserved under Σ-products and closed subsets. It is known that the Sorgenfrey line is almost proximal but not proximal. However, in this paper we show that the Sorgenfrey line is bi-proximal, which shows that our concept of bi-proximal spaces is more general than that of proximal spaces. We then present separation properties of certain bi-proximal spaces and apply them to quasi-uniform box products.</p>

Compact complement topologies and k-spaces

Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.