The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an a...

A topological hyperplane is a subspace of R (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R is a finite set H such that for any nonvoid intersection Y of topological hyperplanes in H and any H ∈ H that intersects but does not contain Y , the intersection is a topological hyperplane in Y . (We also assume...

A b str act . Let Σ be the image of a topological embedding of Sn−2 into Sn. In this paper the homotopy groups of the complement Sn−Σ are studied. In contrast with the situation in the smooth and piecewise linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2. If the first non...

1. INTRODUCTION. For more than twenty years, fractals have intrigued mathematicians and nonmathematicians alike due to their inherent beauty and widespread appearance in nature and computer graphics. Intuitively, a fractal is a geometric object with intricate detail on an arbitrarily small scale and some measure of self-similarity. Formally, a fractal is a metric space with topological dimensio...

Throughout this thesis, we observe close correlations between values of the topological growth rates and various other fractal indices. These observations are based on both analytic derivations and numerical computations of the relevant exponents. In this chapter we derive inequalities that relate our topological growth rates to existing scaling indices such as the box-counting dimension and th...

There is a class of electronic liquids in dimensions greater than one, which show all essential properties of one dimensional electronic physics. These are topological liquids correlated electronic systems with a spectral flow. Compressible topological electronic liquids are superfluids. In this paper we present a study of a conventional model of a topological superfluid in two spatial dimensio...

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function l : π1(X) −→ R+ ∪ ∞ (the value ∞ being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset Conv(X), which is the union of all non-constant minimal...

An exhaustive classification scheme of topological insulators and superconductors is presented. The key property of topological insulators (superconductors) is the appearance of gapless degrees of freedom at the interface/boundary between a topologically trivial and a topologically non-trivial state. Our approach consists in reducing the problem of classifying topological insulators (supercondu...

(i) If the (covering) dimension n of the point space ~ is finite, then ~ is a generalized manifold, and N has the same cohomology as one of the four classical point spaces P2 N, P2t~, P2]H or P2 O, and thus n e {2, 4, 8, 16}, cp. L6wen [111. This holds in particular, if the point rows and the pencils of lines are topological manifolds. In this case, the point rows and pencils of lines are n/2-s...

For non-metrizable spaces the classical Hausdorff dimension is meaningless. We extend the notion of Hausdorff dimension to arbitrary locally convex linear topological spaces, and thus to a large class of non-metrizable spaces. This involves a limiting procedure using the canonical bornological structure. In the case of normed spaces the new notion of Hausdorff dimension is equivalent to the cla...