Some geometric and dynamical properties of the Urysohn space

Melleray, Julien

doi:10.1016/j.topol.2007.04.029

Cited by 42 publications

(59 citation statements)

References 20 publications

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“…This fact shall be applied in the sequel. The reader interested in 'Urysohn-like' spaces is referred to a survey article [10] where one can find more references dealing with this topic.…”

Section: Urysohn Universal Space As a Ntpv Boolean Groupmentioning

confidence: 99%

Normed Topological Pseudovector Groups

Niemiec

2010

Appl Categor Struct

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A normed topological pseudovector group (NTPVG for short) is a valued topological group (V, +, · ) (not necessarily Abelian) endowed with a continuous scalar multiplicationt · x = t x for each t, s ∈ R + and x, y ∈ V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.Keywords Normed vector spaces · Topological groups · Locally compact groups · Valued groups · Dynamical systems · Urysohn universal metric space Mathematics Subject Classifications (2010) 22A99 · 46B99 · 46A99Normed vector spaces are metrizable topological groups of a very special kind. One may say that these are groups 'modelled' on the additive group of real numbers. The possibility of multiplying elements of a group by reals in a homomorphic way and the convex structure of a group facilitate investigations of such groups and has many deep topological and geometric consequences. For example, every normed vector space is an absolute extensor for metric spaces ([5]; see also [2, Theorem II.3.1]), locally compact normed vector spaces are finite dimensional and are uniquely determined (up to topological group isomorphism) by their dimension, non-locally compact completely metrizable normed spaces (that is, infinitedimensional Banach spaces [9]) as topological spaces are also uniquely determined P. Niemiec (B)

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“…This fact shall be applied in the sequel. The reader interested in 'Urysohn-like' spaces is referred to a survey article [10] where one can find more references dealing with this topic.…”

Section: Urysohn Universal Space As a Ntpv Boolean Groupmentioning

confidence: 99%

Normed Topological Pseudovector Groups

Niemiec

2010

Appl Categor Struct

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“…For instance, it is easy to show that (U, δ U ) is compact homogeneous, namely, any isomorphic compact subdiversities are automorphic. This follows the construction in Melleray [8,Section 4.5] for the metric case. It would be worthwhile to study the isometry group of (U, δ U ) along the lines of the results surveyed in [8,Section 4.5].…”

Section: Questionsmentioning

confidence: 82%

“…An example of a complete separable ultrahomogeneous space is the separable in nitedimensional Hilbert space ; see Melleray [8]. Urysohn established that the Urysohn space is the unique (up to isometry) separable metric space satisfying both universality and ultrahomogeneity [8].Here we construct the analogue of the Urysohn space for diversities, a generalization of the concept of metric spaces wherein all nite subsets, and not just pairs of points, are assigned a non-negative value. A diversity is a pair (X, δ) where X is a set and δ is a function from the nite subsets of X to R satisfying …”

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confidence: 99%

A Universal Separable Diversity

Bryant

Nies

Tupper

2017

Analysis and Geometry in Metric Spaces

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Abstract:The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every nite isometry between two nite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any nite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points. We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.Keywords: Diversities, Urysohn space, Katětov functions, universality, ultrahomogeneity MSC: 51F99, 54E50, 54E99 IntroductionUrysohn [10] in 1927 constructed a remarkable metric space which is now named after him. The Urysohn space is the unique (up to isometry) separable complete metric space with the following two properties: (a) universality: all separable metric spaces can be isometrically embedded within it; (b) ultrahomogeneity: any isometry between two nite subspaces of the Urysohn space can be extended to an auto-isometry of the whole space.The property of universality is straightforward to grasp, and holds for several separable complete metric spaces, such as C [ , ]. The property of ultrahomogeneity is less known. Recall that homogeneity of a metric space means that given any two points x, y in the space, there is an automorphism (or self-isometry) of the space that maps x to y. Likewise, a space is 2-homogeneous if for every pair of pairs (x , x ) and (y , y ) such that d(x , x ) = d(y , y ), there is an automorphism of the space taking x to y and x to y . For any k ≥ , k-homogeneity is de ned similarly. Ultrahomogeneity just extends this property to any pair of isometric nite subsets of the space. An example of a complete separable ultrahomogeneous space is the separable in nitedimensional Hilbert space ; see Melleray [8]. Urysohn established that the Urysohn space is the unique (up to isometry) separable metric space satisfying both universality and ultrahomogeneity [8].Here we construct the analogue of the Urysohn space for diversities, a generalization of the concept of metric spaces wherein all nite subsets, and not just pairs of points, are assigned a non-negative value. A diversity is a pair (X, δ) where X is a set and δ is a function from the nite subsets of X to R satisfying that δ is monotonic: A ⊆ B implies δ(A) ≤ δ(B). Also δ is subadditive on sets with nonempty intersection: δ(A ∪ B) ≤ δ(A) + δ(B) whenWe say that a diversity (X, δ) is complete if its induced metric (X, d) is complete [9], and that a diversity is separable if its induced metric is separable.Our main goal is to construct the diversity analog (U, δ U ) of the Urysohn metric space. It is determined uniquely by...

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“…Cameron and Vershik [1] have shown, using different method, that the uniform topology of the group Iso(U) of isometries of the unbounded Urysohn space is nondiscrete as well (see also [4,Exercise 16]). In the next section we shall see that our above example also works in the unbounded case and therefore the group Iso(U) contains nontrivial paths.…”

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confidence: 99%