Related papers: Sharper changes in topologies

Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a free…

Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for Boolean G-actions. We…

A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…

We show that if a (locally compact) group $G$ acts properly on a locally compact $\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is…

We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order…

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $\alpha(X)$ of compact families of open subsets of a…

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\Z$-actions, are coanalytic non-Borel inside…

We prove that in Borel models of arithmetic on an uncountable Polish space, neither addition nor multiplication is continuous. This is an analogue of Tennenbaum's Theorem for topological models of arithmetic. This answers a question of…

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action…

We show that every abelian Polish group is the topological factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced…

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We show that, under mild conditions on a…

Let $G$ be a locally compact group. For every $G$-flow $X$, one can consider the stabilizer map $x \mapsto G_x$, from $X$ to the space $\mathrm{Sub}(G)$ of closed subgroups of $G$. This map is not continuous in general. We prove that if one…

In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which…

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on…

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid $\operatorname{End}(\mathbb{A})$ of a countable relational structure $\mathbb{A}$. As applications, we show…

We define some coding of Borel sets in admissible sets. Using this we generalize certain results from model theory involving admissible sets to the case of continuous actions of closed permutation groups on Polish spaces. In particular we…

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…

For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $\sigma$-algebra of $X$. For $G…

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown…

The set of increasing functions on the rational numbers, equipped with the composition operation, naturally forms a topological semigroup with respect to the topology of pointwise convergence in which a sequence of increasing functions…