A new look at Counterexamples in topology

The book Counterexamples in Topology is a useful catalogue of topological spaces and properties. This thesis extends that catalogue to the properties of sobriety and packedness, and describes some related theory. A purely topological account of sobriety and sober reflections is given, together with an account of the connection with point-free topology which motivates it. Concrete constructions of the sober, T0 and T1 reflections of a topological space are given, and these are calculated for each space in Counterexamples in Topology. These are used to study the relationship between sobriety and the T1 separation property. The notion of a specialization topology is introduced as a means of constructing topological spaces from quasiordered sets. The Alexandrov, Scott and weak topologies are described and shown to be examples of this notion. The sobriety and sober reflections of specialization topologies are considered, and these motivate a suggestion for a generalization of the notion of a topological space. The calculations in this thesis are summarized in two reference tables

On the Alexandrov Topology of sub-Lorentzian Manifolds

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.Comment: 20 page

Topological Schemas of Memory Spaces

Hippocampal cognitive map---a neuronal representation of the spatial environment---is broadly discussed in the computational neuroscience literature for decades. More recent studies point out that hippocampus plays a major role in producing yet another cognitive framework that incorporates not only spatial, but also nonspatial memories---the memory space. However, unlike cognitive maps, memory spaces have been barely studied from a theoretical perspective. Here we propose an approach for modeling hippocampal memory spaces as an epiphenomenon of neuronal spiking activity. First, we suggest that the memory space may be viewed as a finite topological space---a hypothesis that allows treating both spatial and nonspatial aspects of hippocampal function on equal footing. We then model the topological properties of the memory space to demonstrate that this concept naturally incorporates the notion of a cognitive map. Lastly, we suggest a formal description of the memory consolidation process and point out a connection between the proposed model of the memory spaces to the so-called Morris' schemas, which emerge as the most compact representation of the memory structure.Comment: 24 pages, 8 Figures, 1 Suppl. Figur

C*-Algebras over Topological Spaces: The Bootstrap Class

We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space. We introduce and describe an analogue of the bootstrap class for C*-algebras over a finite topological space.Comment: Final version, very minor change