Metric Spaces Research Articles

Almost Finiteness and Groups of Dynamical Origin

Abstract We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.

The space of traces of the free group and free products of matrix algebras

We show that the space of traces of free products of the form C(X1)⁎C(X2), where X1 and X2 are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group Fd on 2≤d≤∞ generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras Mn(C)⁎Mn(C) is a Poulsen simplex as well, answering a question of Musat and Rørdam for n≥4. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

FIXED POINT THEOREMS IN SOME FUZZY METRIC SPACES VIA INTERPOLATIVE CONTRACTIONS

In this article, an interpolative contraction existing in the literature is adapted to different fuzzy metric spaces. Using this contraction, a fixed point theorem in two fuzzy metric spaces is proven and an example is presented. Thus, a more general form of some concepts and theorems existing in the literature has been obtained.

Non-stationary Itô–Kawada and ergodic theorems for random isometries

We consider a non-stationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic image, we establish a weak-* convergence to the unique invariant under isometries measure, ergodic theorem and large deviation type estimate. We also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, we prove a non-stationary analog of classical Itô–Kawada theorem and give a new alternative proof for the stationary case.

Compact elements and the hypocompact radical of crossed products

Abstract We characterize the compact elements and the hypocompact radical of a crossed product $C_0(X)\times _\phi \mathbb Z$ , where X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a homeomorphism, in terms of the corresponding dynamical system $(X,\phi )$ .

Reflecting topological properties in closed separable subspaces

We establish that, for any metrizable space X, if every closed separable subspace of X has the Baire property, then the space X has the Baire property. The same reflection result holds for spaces Cp(X). It is also shown that all closed separable subspaces of Cp(X) are Čech-complete if and only if X is a functionally countable P-space. If A¯ is pseudocomplete for every countable set A⊂Cp(X), then Cp(X) is pseudocomplete. To show that zero-dimensionality does not reflect in closed separable subspaces of metrizable spaces, under Jensen’s Axiom ♢, we give an example of a connected metrizable space M such that A¯ is countable and hence zero-dimensional for every countable set A⊂M.

Rational gluing in edge replacement systems

In this paper, we prove the rationality of the gluing relation of edge replacement systems, which were introduced for studying rearrangement groups of fractals. More precisely, we describe an algorithmic procedure for building a finite state automaton that recognizes pairs of equivalent sequences that are glued in the fractal. This fits in recent interest toward the rationality of gluing relations on totally disconnected compact metrizable spaces.

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

On the Borel complexity and the complete metrizability of spaces of metrics

Abstract Given a metrizable space X X , let A M ( X ) AM\left(X) be the space of continuous bounded admissible metrics on X X , which is endowed with the sup-metric. In this article, we shall investigate the Borel complexity and the complete metrizability of A M ( X ) AM\left(X) and show that a separable metrizable space X X is σ \sigma -compact if and only if A M ( X ) AM\left(X) is completely metrizable.

THE SHAPE OF COMPACT COVERS

Abstract For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ , and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa. We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces. Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $ -pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$ .

Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.

A note on fibers and Vietoris topologies of paratopological groups

If f: G → Y is an irreducible closed continuous mapping defined on a regular weakly collectionwise normal first-countable meta-Lindelöf locally σ paratopological group G onto a T 1-space Y which has a neighborhood ωω -base at a point y ∈ Y, then f −1(y) is σ-compact in G. We prove that if a Fréchet-Urysohn space X has strong α 4 -property and a weakly countably complete base , then X is first-countable, where M is a separable and metrizable space and = {F: F is a non-empty compact subset of M } and with the Vietoris topology. By this result we can get the first-countability of certain paratopological groups.

Every connected first countable T1-space is a continuous open image of a connected metrizable space

Every connected first countable T1-space is a continuous open image of a connected metrizable space

Lipschitz-free spaces over properly metrizable spaces and approximation properties

Let T T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let M T \mathcal {M}^T be the non-empty set of all proper metrics d d on T T compatible with its topology, and equip M T \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T 2 T^2 . We prove that the set A T , 1 \mathcal {A}^{T,1} of metrics d ∈ M T d\in \mathcal {M}^T for which the Lipschitz-free space F ( T , d ) \mathcal {F}(T,d) has the metric approximation property is a dense set in M T \mathcal {M}^T , and is furthermore residual in M T \mathcal {M}^T when T T is zero-dimensional. We also prove that if T T is uncountable then the set A f T \mathcal {A}^T_f of metrics d ∈ M T d\in \mathcal {M}^T for which F ( T , d ) \mathcal {F}(T,d) fails the approximation property is dense in M T \mathcal {M}^T . Combining the last statement with a result of Dalet, we conclude that for any ‘properly metrizable’ space T T , A f T \mathcal {A}^T_f is either empty or dense in M T \mathcal {M}^T .

Compact subspaces of the space of separately continuous functions with the cross-uniform topology

We consider two natural topologies on the space S(X×Y,Z) of all separately continuous functions defined on the product of two topological spaces X and Y and ranged into a topological or metric space Z. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if X and Y are pseudocompacts and Z is a metric space. We prove that a compact space K embeds into S(X×Y,Z) for infinite compacts X, Y and a metrizable space Z⊇R if and only if the weight of K is less than the sharp cellularity of both spaces X and Y.

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

AbstractIn this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite‐by‐virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property and ‐stability for their crossed product ‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

Lebesgue number and total boundedness

A generalization of the Lebesgue number lemma is obtained. For a metric space X in the class of strongly metrizable spaces, sufficient conditions for each open cover of X with a Lebesgue number has a finite subcover are obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space X has a Lebesgue number, then X is totally bounded. A property for metric spaces which is a generalization of connectedness and Menger convexity is introduced. It is observed that Atsujiness and compactness are equivalent for a metric space with this introduced property as well as for a chainable metric space.

Actions of finitely generated groups on compact metric spaces

Let Γ \Gamma be a finitely generated group which admits an action by homeomorphisms on a metrizable space X X . We show that there is a metric on X X defining the original topology such that for this metric, the action is by bi-Lipschitz transformations.

Compact spaces homeomorphic to their respective squares

We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of Włodzimierz J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.