We prove an analogue of Voiculescu’s theorem: Relative bicommutant of a separable unital subalgebra A of an ultraproduct of simple unital C*-algebras is equal to A. Ultrapowers1 AU of separable algebras are, being subject to well-developed model-theoretic methods, reasonably well-understood (see e.g. [9, Theorem 1.2] and §2). Since the early 1970s and the influential work of McDuff and Connes c...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

in this paper, we will give necessary and sufficient conditions for certain hnnextensions of subgroup separable groups with normal associated subgroup to be conjugacyseparable. in fact, we will show that these hnn extensions are conjugacy separable if and onlyif the normalizer of one of its associated subgroup is conjugacy separable.

We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably h...

In this article, we study two open problems posed by Edelman & Reiner about topology of certain simplicial complexes associated with abstract convex geometries. As a special case, we concentrate on generalized convex shellings, which were introduced by Kashiwabara, Nakamura & Okamoto for their representation theorem of abstract convex geometries. We answer one of the problems affirmatively for ...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional on the space of Lipschitz maps into the domain of non-empty, convex and weak*-compact valued functions ordered by reverse inclusion, is continuous. For finite dime...

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits utility representation. particular, show that, for completely regular spaces, topology is useful, if and only it separable, isolated chain open closed sets countable. As specific application optimization theory, characteri...

Preterm birth is associated with altered connectivity of neural circuits. We developed a tract segmentation method that provides measures of tract shape and integrity (probabilistic neighborhood tractography, PNT) from diffusion MRI (dMRI) data to test the hypotheses: 1) preterm birth is associated with alterations in tract topology (R), and tract-averaged mean diffusivity (〈D〉) and fractional ...

Reflection theorems in set-theoretic topology typically take the following form: if all “small” subspaces of a suitable kind of space X have a property P, then so does the whole space. Here “small” often means “of cardinality א1” but in Sections 1 and 2 of this paper it will mean “separable”. Usually, but not always, some sort of large cardinal axiom is needed, and the reflection theorem holds ...

In this paper we show that C-k(X), the set of continuous, realvalued functions on X topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in Rx even when X is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces X for which C-n(X) lies nowhere in the projective hierarchy of the co...