相关论文: Stationary Sets in Topological and Paratopological…

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…

We study topological groups of monotonic autohomeomorphisms on a generalized ordered space $L$. We find a condition that is necessary and sufficient for the set of all monotonic autohomeomorphisms on $L$ along with the function composition…

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm…

Given a set $X$ and a family $G$ of self-maps of $X$, we study the problem of the existence of a non-discrete Hausdorff topology on $X$ with respect to which all functions $f\in G$ are continuous. A topology on $X$ with this property is…

In this paper, we extend the concept of generalized entropy to uniform spaces, allowing computations beyond metrizable settings. We apply this to parabolic dynamics - systems with a unique fixed point uniformly attracting all compact…

Let a group $\Gamma$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural…

We show that (in ZFC) every infinite set S can be equipped with 2^|S| complete metrics which generate mutually non-homeomorphic scattered order topologies on S. Furthermore, we show that (in ZFC) every uncountable set S can be equipped with…

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it…

For any countable group, and also for any locally compact second countable, compactly generated topological group, G, we show the existence of a "universal" hypercyclic (i.e. topologically transitive) representation on a Hilbert space, in…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

The aim of this note is to prove that the set of proper normal subgroups of a group endowed with coarse lower topology is a spectral space.

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $\kappa$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both…

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent…

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular…

Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a countable base and is not countable, then T has cardinality at least continuum.

We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread.…

Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser is open in G. We begin an investigation of the structure of the family of closed locally normal subgroups of G. Modulo…

We show that it is consistent that for some uncountable cardinal k, all compactifications of the countable discrete space with remainders homeomorphic to $D^k$ are homeomorphic to each other. On the other hand, there are $2^c$ pairwise…

Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the…

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…