Let M be a closed connected manifold and let D be an elliptic operator on M . Let G be a discrete countable group and let M̃ → M be a principal G-bundle. Connes and Moscovici showed that this data defines an analytic index ind`1(G)(D) ∈ K0(`(G)). If B is a unital tracial C*-algebra, we give a formula for the trace of the image of ind`1(G)(D) in K0(B) under the map induced by a quasi-representati...

Abstract Variants of Kleene algebra have been used to provide foundations reasoning about programs, for instance by representing Hoare Logic (HL) in algebra. That work has generally emphasised program correctness, i.e., proving the absence bugs. Recently, Incorrectness (IL) advanced as a formalism dual problem: presence IL is intended underpin use logic testing and static bug finding. Here, we ...

A topological space has calibre ω1 (resp., calibre (ω1,ω)) if every point-countable (resp., point-finite) collection of nonempty open sets is countable. It has compact-calibre ω1 (resp., compact-calibre (ω1,ω)) if, for every family of uncountably many nonempty open sets, there is some compact set which meets uncountably many (resp., infinitely many) of them. It has CCC (resp., DCCC) if every di...

Some results are obtained concerning n(T), the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links...

We introduce a notion of L-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L-Betti numbers of countable discrete groups for lattices. In this way, several new computations are obtained for countable groups, including lattices in algebraic groups over local elds, and Kac-Moody lattices. We also extend the vanishing of reduce...

No countable C,-free graph contains every countable C,-free graph as a subgraph, for n 2 4. For n = 4, this was proved earlier by J. Pach.

Given a space 〈X,J 〉 in an elementary submodel M of H(θ), define XM to be X ∩ M with the topology generated by {U ∩ M : U ∈ J ∩ M}. It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X = XM . Assuming CH+SCH (the Singular Cardinals Hypothesis...

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it,...

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

We give, for each countable ordinal ξ ≥ 1, an example of a ∆2 countable union of Borel rectangles that cannot be decomposed into countably many Πξ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ∆ξ-measurable countable coloring. 2010 Mathematics Subject Classification. Primary: 03E15, Secon...