Strongly Rigid Metrics and Zero Dimensionality (original) (raw)

Linearly rigid metric spaces and the embedding problem

2006

We consider the problem of isometric embedding of the metric spaces to the Banach spaces; and introduce and study the remarkable class of so called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The various properties of linearly rigid spaces and related spaces are considered.

The existence of rigid measurable spaces

Topology and its Applications, 1989

For each uncountable cardinal K we construct 2" dense unbounded chains (S,, <) of cardinality K which as topological spaces (endowed with the order-topology) have, in particular, the following properties: They are each O-dimensional and mono-rigid, i.e. the only embedding of .S, into itself is the identity, and they are pairwise nonembeddable into each other. If K 2 KZ, the sets S, can be chosen such that, in addition, each G,-set is open and hence the measurable spaces (S,, gr), where 3, is the c-algebra of all clopen subsets of S,, are mono-rigid and pairwise nonembeddable into each other.

On metric spaces with the properties of de Groot and Nagata in dimension one

Topology and its Applications, 2010

A metric space (X, d) has the de Groot property GPn if for any points x 0 , x 1 ,. .. , x n+2 ∈ X there are positive indices i, j, k ≤ n + 2 such that i = j and d(x i , x j) ≤ d(x 0 , x k). If, in addition, k ∈ {i, j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X) ≤ n iff X has an admissible GPn-metric iff X has an admissible NPn-metric. We prove that an embedding f : (0, 1) → X of the interval (0, 1) ⊂ R into a locally connected metric space X with property GP 1 (resp. NP 1) is open, provided f is an isometric embedding (resp. f has distortion Dist(f) = f Lip • f −1 Lip < 2). This implies that the Euclidean metric cannot be extended from the interval [−1, 1] to an admissible GP 1-metric on the triode T = [−1, 1]∪ [0, i]. Another corollary says that a topologically homogeneous GP 1-space cannot contain an isometric copy of the interval (0, 1) and a topological copy of the triode T simultaneously. Also we prove that a GP 1-metric space X containing an isometric copy of each compact NP 1-metric space has density ≥ c.

Rigidity of hyperspaces

Rocky Mountain Journal of Mathematics, 2015

Given a metric continuum X, we consider the following hyperspaces of X: 2 X , Cn(X) and Fn(X) (n ∈ N). Let F 1 (X) = {{x} : x ∈ X}. A hyperspace K(X) of X is said to be rigid, provided that for every homeomorphism h : K(X) → K(X), we have h(F 1 (X)) = F 1 (X). In this paper, we study conditions under which a continuum X has a rigid hyperspace Cn(X). Among others, we consider families of continua, such as dendroids, Peano continua, hereditarily indecomposable continua and smooth fans.

How rigid the finite ultrametric spaces can be?

Journal of Fixed Point Theory and Applications, 2016

A metric space X is rigid if the isometry group of X is trivial. The finite ultrametric spaces X with |X| ≥ 2 are not rigid since for every such X there is a self-isometry having exactly |X|−2 fixed points. Using the representing trees we characterize the finite ultrametric spaces X for which every self-isometry has at least |X| − 2 fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.

On the Structure of Metric-like Spaces

The main purpose of this paper is to introduce several concepts of the metric-like spaces. For instance, we define concepts such as equal-like points, cluster points and completely separate points. Furthermore, this paper is an attempt to present compatibility definitions for the distance between a point and a subset of a metric-like space and also for the distance between two subsets of a metric-like space. In this study, we define the diameter of a subset of a metric-like space, and then we provide a definition for bounded subsets of a metric-like space. In line with the aforemen-tioned issues, various examples are provided to better understand this space.

Four metric conditions characterizing Čech dimension zero

Proceedings of the American Mathematical Society, 1977

If (X,d) is a metric space let dx(y) = d(x,y). It is proved that if each x in A-has a neighbourhood P with dx(P) not dense in any neighbourhood of 0 in [0,oo) then Ind X = 0. This metric condition characterizes metrizable spaces which have Cech dimension zero. Three other metric characterizations are given.

Some remarks on the metrizability of some metric-like structures

Carpathian Journal of Mathematics, 2021

The main purpose of this article is to provide alternative proofs of the metrizability of metric-like spaces like b-metric spaces, \mathcal{F}-metric spaces, and \theta-metric spaces. We improve upon the metrizability result of An et al. [Topology Appl. 185–186 (2015)] for b-metric spaces. Moreover, we provide two shorter proofs of the metrizability of \mathcal{F}-metric spaces, recently introduced by Jleli and Samet. Furthermore, we give a partial answer to an open problem regarding the openness of \mathcal{F}-open balls in \mathcal{F}-metric spaces. Finally, we give an alternative proof of the metrizability of \theta-metric spaces.

Quasi-isometric rigidity in low dimensional topology

The early work of Mostow, Margulis and Prasad on rigidity of arithmetic lattices has evolved into a broad use of quasi-isometry techniques in group theory and low dimensional topology. The word metric on a finitely generated group makes it into a metric space which is uniquely determined up to the geometric relation called quasi-isometry, despite the fact that the metric depends on the choice of generating set. As for lattices in suitable Lie groups, where quasi-isometry of lattices implies commensurability, the general quasi-isometric ...

Some Bernstein-type rigidity theorems

Journal of Mathematical Analysis and Applications, 2012

We prove some Bernstein-type rigidity theorems for complete submanifolds in a Euclidean space and space-like submanifolds of a Lorentzian space. In particular, we obtain a Bernstein rigidity theorem for complete minimal submanifolds of arbitrary codimension in Euclidean space.