Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0701172

On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

Antonios Manoussos·Polychronis Strantzalos
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action $(G,X)$ we construct a zero-dimensional compactification $μX$ of $X$ with the properties: (a) there exists an extension of the action on $μX$, (b) if $μL\subseteq μX\setminus X$ is the set of the limit points of the orbits of the initial action in $μX$, then the restricted action $(G,μX\setminus μL)$ remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on $μX\setminus μL$ by that of $μX$, and (c) $μX$ is the maximal among the zero-dimensional compactifications of $X$ with these properties. Proper actions are usually embedded in the end point compactification $εX$ of $X$, in order to obtain topological invariants concerning the cardinality of the space of the ends of $X$, provided that $X$ has an additional "nice" property of rather local character ("property Z", i.e., every compact subset of $X$ is contained in a compact and connected one). If the considered space has this property, our new compactification coincides with the end point one. On the other hand, we give an example of a space not having the "property Z" for which our compactification is different from the end point compactification. As an application, we show that the invariant concerning the cardinality of the ends of $X$ holds also for a class of actions strictly containing the properly discontinuous ones and for spaces not necessarily having "property Z".

Explore related subjects

math.GNmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Antonios Manoussos, Polychronis Strantzalos. 2007-01-05. On embeddings of proper and equicontinuous actions in zero-dimensional compactifications. https://arxiv.org/abs/math/0701172

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Borelness of Moduli Spaces of Metrics Implies Separability

Let $X$ be a metrizable space, and let $\mathrm{Met}(X)$ denote the space of metrics compatible with the topology of $X$, regarded as a subspace of the space of continuous pseudometrics with the topology induced by the supremum metric. We first prove that if $Z$ is a discrete space whose cardinality is the first uncountable cardinal, then $\mathrm{Met}(Z)$ is not Borel. As a consequence, if $\mathrm{Met}(X)$ is Borel, then $X$ is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space $X$ is completely metrizable if and only if $X$ is $σ$-compact. We also establish non-Archimedean analogues for spaces of ultrametrics. We further prove componentwise versions of these results, where two metrics belong to the same component if their distance is finite.

math.GN↗
arXiv2026

Four problems about the irreducible determined $T_0$-topologies

We answer four questions on irreducible determined $T_0$-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\ 382 (2026), 109762). For every $T_0$ topology $τ$, the family $τ_r$ used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: $τ_r=\rd(τ)=\Rd(τ)$. For an arbitrary poset $P$, we describe the weakest order-compatible irreducible determined topology $ρ(P)$ using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe $ρ(P)$-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common $T_1$ example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.

math.GN↗
arXiv2026

Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups

A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a $k$-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if $X$ is a separable Baire space, $Y$ is a feebly compact space and $Z$ is a regular space, then every separately continuous map $X\times Y\to Z$ is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is $Γ^{BM}_o$-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly $p$-spaces; feebly $p$-spaces form a wider class of spaces, which contains all feebly compact spaces and all $p$-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.

math.GN↗