Related papers: A Note on Locally Compact Subsemigroups of Compact…

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy…

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this…

In this short note we prove that a group G is lcH-slender -- that is, every abstract group homomorphism from a locally compact Hausdorff topological group to G has an open kernel -- if and only if G is torsion-free and does not include Q or…

Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action,…

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their…

If a finitely generated semigroup S has a hopfian (meaning: every surjective endomorphism is an automorphism) cofinite subsemigroup T then S is hopfian too. This no longer holds if S is not finitely generated. There exists a finitely…

The left multiplicative continuous compactification of a semitopological semigroup is the universal semigroup compactification. In this paper an internal construction of a semigroup compactification of a semitopological semigroup is…

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is…

A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative…

A space $X$ is submaximal if any dense subset of $X$ is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly $\sigma$-discrete. Moreover, we prove that every submaximal strongly…

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of…

We show that if a subsemigroup $S$ of the bicyclic monoid ${\mathscr{C}}(p,q)$ contains infinitely many idempotents then $S$ admits only the discrete Hausdorff shift-continuous topology. Also we proof that every right-continuous…

This note is devoted to proving the following result: given a compact metrizable group G, there is a compact metric space K such that G is isomorphic (as a topological group) to the isometry group of K.

By Cartan's Theorem, every closed subgroup $H$ of a real (or $p$-adic) Lie group $G$ is a Lie subgroup. For Lie groups over a local field ${\mathbb K}$ of positive characteristic, the analogous conclusion is known to be wrong. We show more:…

We introduce the concept of a topological J-group and determine for many important examples of topological groups if they are topological J-groups or not. Besides other results, we show that the underlying topological space of a pathwise…

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for…

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only…

Are all subcategories of locally finitely presentable categories that are closed under limits and $\lambda$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the…

The scale of an endomorphism, $\alpha$, of a totally disconnected, locally compact group $G$ is the minimum index $[\alpha(U) : \alpha(U)\cap U]$, for $U$ a compact, open subgroup of $G$. A structural characterization of subgroups at which…

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are:…