Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.     

Weighted mean topological dimension

This paper is devoted to the investigation of weighted mean topological dimension in dynamical systems. We show that the weighted mean dimension is not larger than the weighted metric mean dimension, which generalizes the classical result of Lindenstrauss and Weiss [16]. We also establish the relationship between the weighted mean dimension and the weighted topological entropy of dynamical systems, which indicates that each system with finite weighted topological entropy or small boundary property has zero weighted mean dimension.     

Covering dimension and finite spaces

Finite topological spaces, that is spaces with a finite number of points, have a wide range of applications in many areas such as computer graphics and image analysis. In this paper we study the covering dimension of a finite topological space. In particular, we give an algorithm for computing the covering dimension of a finite topological space using matrix algebra.     

Covering dimension and finite-to-one maps

Hurewicz characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most n²-to-1 map.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Covering a graph by topological complete subgraphs

Every graph onn vertices can be covered byex(n, TK ₄ ) =2n – 3 TK ₄ -subgraphs and edges. A similar result might hold for arbitrary topological (complete) subgraphs as indicated by the following result: Every graph onn vertices can be covered by 65ex(n, TK _p )TK _p -subgraphs and edges. IfH is a connected graph (|H| 3) then every graph onn vertices can be covered by 400ex(n, TH) TH-subgraphs and edges. Similar questions for coverings by odd and even cycles are also considered.This author's work was done while he was a guest at the Hungarian Academy of Sciences.     

Covering dimension d by a normal base

S.D. Iliadis introduced the concept of the dimension-like functions of type dim using the notion of a normal base. Since he considered this notion only from the point of view of the existence of universal spaces in different classes of spaces, in the present paper the basic theorems of dimension theory are obtained for these functions. Their relationship with the relative dimensions of A. Chigogidze and the uniform dimensions of M. Charalambous are shown.     

Covering groups of nonconnected topological groups and 2-groups

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are equal.     

Canonical subbase-compactness of topological products

For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory.Results: (1) Products of finite spaces are canonically subbase-compact iff AC(fin), the axiom of choice for finite sets, holds.(2) Products of n-element spaces are canonically subbase-compact iff AC(<n), the axiom of choice for sets with less than n elements, holds.(3) Products of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds.(4) All powers XI of a compact space X are canonically subbase compact iff X is a Loeb-space.These results imply that in ZF the implications

     

The topological dimension of limits of vertex replacements

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure, a sequence of graphs {Rn(G)} is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.     

Minimal systems of arbitrary large mean topological dimension

Let G be a countable amenable group and P a polyhedron. The mean topological dimension mdim(X,G) of a subshift X ⊂ P ^G is a real number satisfying 0 ≤ mdim(X,G) ≤ dim(P), where dim(P) denotes the usual topological dimension of P. We give a construction of minimal subshifts X ⊂ P ^G with mean topological dimension arbitrarily close to dim(P).     

Some aspects of dimension theory for topological groups

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of $[0,1]$. And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let $G$ be a topological group with a countable network. Does it follow that $\dim G=\mathrm{ind}G=\mathrm{Ind}G$? The same question if $X$ is a compact coset space. We also do not know whether the inequality $\dim (G\times H)\le \dim G+\dim H$ holds for arbitrary topological groups $G$ and $H$ which are subgroups of $\sigma$-compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.     

Noetherian type in topological products

The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2:

1. There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}.
2. In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace.

The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\) , is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\) . We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an (?₄, ?₁)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ?₄ on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\) . The proof uses a method that was introduced by Shelah in 1993 [33].     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .

     

Invariants and dimension functions in general topological spaces

Summary We give here a general method of defining invariants in topological spaces. As applications we give a new concept of dimension and a theorem on separation of spaces by sub-spaces. Entrata in Redazione il 6 luglio 1968.     

On rectangular products of topological spaces

Several results on rectangular products in the sense of B.A. Pasynkov will be obtained, one of which asserts that for a Tychonoff space X, X × Y is rectangular for any space Y iff X is locally compact and paracompact.