Two characterizations of topological spaces with no infinite discrete subspace

We give two characteristic properties of topological spaces with no infinite discrete subspaces. The first one was obtained recently by the first author. The full result extends well-known characterizations of posets with no infinite antichain.Comment: 10 pages, no figures Replace second author by first author in the abstract and in the first line after Theorem

Some closure operations in Zariski-Riemann spaces of valuation domains: a survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains

Borel density for approximate lattices

We extend classical density theorems of Borel and Dani--Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.Comment: 17 pages, 0 figures. Comments are welcome

Topologically subordered rectifiable spaces and compactifications

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$ we have $\phi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we mainly discuss the rectifiable spaces which are suborderable, and show that if a rectifiable space is suborderable then it is metrizable or a totally disconnected P-space, which improves a theorem of A.V. Arhangel'ski\v\i\ in \cite{A20092}. As an applications, we discuss the remainders of the Hausdorff compactifications of GO-spaces which are rectifiable, and we mainly concerned with the following statement, and under what condition $\Phi$ it is true. Statement: Suppose that $G$ is a non-locally compact GO-space which is rectifiable, and that $Y=bG\setminus G$ has (locally) a property-$\Phi$. Then $G$ and $bG$ are separable and metrizable. Moreover, we also consieder some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.Comment: 14 pages (replace

Normality in terms of distances and contractions

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of $\gamma$-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of $\gamma$-separated sets by some Urysohn contractive map, (2) to Kat\v{e}tov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space. The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Kat\v{e}tov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered

Local models of Shimura varieties, I. Geometry and combinatorics

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the "Handbook of Moduli