There exists a physically well motivated method for approximating manifolds by certain topological spaces with a finite or a countable set of points. These spaces, which are partially ordered sets (posets) have the power to effectively reproduce important topological features of continuum physics like winding numbers and fractional statistics, and that too often with just a few points. In this ...

Let G = (V,E) be a locally finite graph, i.e. a graph in which every vertex has finitely many adjacent vertices. In this paper, we associate a topology to G, called graphic topology of G and we show that it is an Alexandroff topology, i.e. a topology in which intersec- tion of every family of open sets is open. Then we investigate some properties of this topology. Our motivation is to give an e...

There are dierent ways to code the geodesic flows on surfaces with negative curvature. Such code spaces give a useful tool to verify the dynamical properties of geodesic flows. Here we consider special subspaces of geodesic flows on Hecke surface whose arithmetic codings varies on a set with innite alphabet. Then we will compare the topological complexity of them by computing their topological ...

Let L be a completely distributive lattice and C a topological construct; a process is given in this paper to obtain a topological construct C(L), called the tower extension of C (indexed by L). This process contains the constructions of probabilistic topological spaces, probabilistic pretopological spaces, probabilistic pseudotopological spaces, limit tower spaces, pretopological approach spac...

Let m be a Radon measure on a Hausdorff topological space X. Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered. The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible. For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer...

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which...

This dissertation provides an introduction to the ideas employed in topological quantum field theory. We illustrate how the field began by considering knot invariants of three-manifolds and demonstrating the consequences of defining such a theory axiomatically. The ideas of category theory are introduced and we show that what we are actually concerned with are symmetric monoidal functors from t...

It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ult...

By using the representational power of Chu spaces we define the notion of a generalized topological space (or GTS, for short), i.e., a mathematical structure that generalizes the notion of a topological space. We demonstrate that these topological spaces have as special cases known topological spaces. Furthermore, we develop the various topological notions and concepts for GTS. Moreover, since ...

We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F , the G-CW -version EF (G) and the numerable G-space version JF (G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if F is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact op...