Related papers: Linear algebra and unification of geometries in al…

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set…

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the…

This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson…

Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their Higson…

We show that under mild set theoretic hypotheses we have rigidity for algebras of continuous functions over Higson coronas, topological spaces arising in coarse geometry. In particular, we show that under $\mathsf{OCA}$ and $\mathsf…

We develop some nonstandard techniques for bornological and coarse spaces. We first generalise the notion of bornology to prebornology, which better fits to coarse spaces. We then give nonstandard characterisations of some basic large-scale…

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly…

In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and…

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the…

We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of $C^*$-algebras and nondegenerate multiplier-valued $*$-homomorphisms. In both situations, we…

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0 they see the…

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse…

Let $X$ and $Y$ be proper metric spaces. We show that a coarsely $n$-to-$1$ map $f\colon X\to Y$ induces an $n$-to-$1$ map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems…

We show that in dimensions $>1$ the cohomology groups of the Higson compactification of the hyperbolic space $\H^n$ with respect to the $C_0$ coarse structure are trivial. Also we prove that the cohomology groups of the Higson…

This paper deepens into the relations between coarse spaces and compactifications, by defining a $C_0$ coarse structure attached to a family of pseudometrics. This definition allow us to give a more topological point of view on the…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

Several mathematicians, including myself, have studied some unifications in general topological spaces as well as in fuzzy topological spaces. For instance in our earlier works, using operations on topological spaces, we have tried to unify…

In this paper, we mainly study the compactness and local structure of immersing surfaces in $\mathbb{R}^n$ with local uniform bounded area and small total curvature $\int_{\Sigma\cap B_1(0)} |A|^2$. A key ingredient is a new quantity which…

The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure,…

A coarse compactification of a proper metric space $X$ is any compactification of $X$ that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of $X$ whose corona is of dimension $0$.…