Non-regular Spaces Research Articles

Generalized cell structures

Cell structures were introduced by W. Debski and E. Tymchatyn as a way to study some classes of topological spaces and their continuous functions through discrete approximations. In this work we weaken the notion of cell structure and prove that, the resulting class of topological space admitting such a generalized cell structure includes non-regular spaces.

Selective versions of θ-density

In [3] the authors initiate the study of selective versions of the notion of θ-separability in non-regular spaces. In this paper we continue this investigation by establishing connections between the familiar cardinal numbers arising in the set theory of the real line, and game-theoretic assertions regarding θ-separability.

Non-Regular Spacetime Geometry

PrefaceUntil recently, mathematical studies of Lorentzian causality theory assumed, explicitly or implicitly, smoothness of the metric. This changed a few years ago, after independent work by Fathi and Siconolfi, and by Chruściel and Grant. Since then many papers began to explore causality under weaker conditions such as C1,1 metrics, Lipschitz metrics, continuous metrics (and cone structures), or even topological settings. Indeed, new insights are needed when the metric is not twice-differentiable. Hence, manifolds for which the metric is not C2 might be called non-regular, though closer inspection reveals that there are not so many differences between the C2 and the C1,1 theories, so one can also refer to the non-regular theories as those whose metric has regularity below C1,1.In Riemannian signature geometers studied non-regular manifolds using comparison geometry and other approaches. Many of these techniques have yet to be exported to the Lorentzian signature since, of course, a number of difficulties have to be addressed. In spite of this, mathematical physicists study non-regular spaces with a strong motivation that stems from the desire of understanding the ultimate nature of spacetime. The idea is that at the quantum level and hence whenever the physical conditions become extreme (say gravitational collapse, or origin of the universe) the spacetime manifold should perhaps not be approximated by a smooth manifold. If this expectation is correct then an understanding of the role of differentiability conditions in general relativity might give hints to the very nature of gravity and spacetime at the quantum scale.

Variations on selective separability in non-regular spaces

We study various variations on selective separability in non-regular topological spaces. We use the notions of θ-closure and θ-density to define selective versions of θ-separability. These properties are also related to topological games.

Dirac operators and the calculation of the Connes metric on arbitrary (infinite) graphs

As an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite) graphs and use it to study spectral properties of graph-Laplacians and graph-Dirac-operators. We define a spectral triplet sharing most of the properties of what Connes calls a spectral triple. With the help of this scheme we derive an explicit expression for the Connes-distance function on general directed or undirected graphs. We derive a series of apriori estimates and calculate it for a variety of examples of graphs. As a possibly interesting aside, we show that the natural setting of approaching such problems may be the framework of (non-)linear programming or optimization. We compare our results (arrived at within our particular framework) with the results of other authors and show that the seeming differences depend on the use of different graph-geometries and/or Dirac operators.