Axioms for Sequential Convergence

Gutierres, Gonçalo; Hofmann, Dirk

doi:10.1007/s10485-007-9095-2

Cited by 5 publications

(6 citation statements)

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“…,Ω ′ i ,g = 0 , we have lim i max{ d(x, ψ i (x)) | x ∈ Ω ′ i ∩ B(x i , r) } = 0 (17) for some r > 0; and (ii) weakly aperiodic if, to get (17), besides the conditions of (i), it is also required that there is some s > 0 and there are points z i ∈ Ω ′ i such that φ ij (z i ) = z j and d(z i , ψ i (z i )) < s. Lemma 12.5. The following properties hold for any complete connected Riemannian n-manifold M :…”

Section: And We Can Canonically Identify πmentioning

confidence: 99%

“…[33, Chapter 10, Section 3Here, a domain in M is a connected C ∞ submanifold, possibly with boundary, of the same dimension as M .It is admitted that C ∞ convergence defines a topology on M * (n) [32]. However we are not aware of any proof in the literature showing that it satisfies the conditions to describe a topology [28], [17] (see also [26] and [27] if C ∞ convergence were defined with nets or filters). This is only proved on subspaces defined by manifolds of equi-bounded geometry, where the C ∞ convergence coincides with convergence in M * [29] (see also [33, Chapter 10]).…”

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A universal Riemannian foliated space

López¹,

Lijó²,

Candel³

2014

Preprint

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It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M ∞ * (n), with the topology described by the C ∞ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace M ∞ * ,lnp (n) ⊂ M ∞ * (n), which becomes a C ∞ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspaceby the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M ∞ * ,lnp (n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M ∞ * ,lnp (n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Foliated spaces 3 2.2. Riemannian geometry 5 3. Quasi-isometries 6 4. Partial quasi-isometries 11 5. The C ∞ topology on M * (n) 12 6. Convergence in the C ∞ topology 13 7. M ∞ * (n) is Polish 15 8. Some basic properties of M ∞ * ,lnp (n) 17 9. Canonical bundles over M ∞ * ,lnp (n) 20 10. Center of mass 24 11. Foliated structure of M ∞ * ,lnp (n) 25 12. Saturated subspaces of M ∞ * ,lnp (n) 29 Acknowledgements 30 References 30 Key words and phrases. C ∞ convergence of Riemannian manifolds; locally non-periodic Riemannian manifolds; Riemannian foliated space. 1 Definition 1.1 (See e.g. [33, Chapter 10, Section 3Here, a domain in M is a connected C ∞ submanifold, possibly with boundary, of the same dimension as M .It is admitted that C ∞ convergence defines a topology on M * (n) [32]. However we are not aware of any proof in the literature showing that it satisfies the conditions to describe a topology [28], [17] (see also [26] and [27] if C ∞ convergence were defined with nets or filters). This is only proved on subspaces defined by manifolds of equi-bounded geometry, where the C ∞ convergence coincides with convergence in M * [29] (see also [33, Chapter 10]). The first main theorem of the paper is the following.Recall that a space is called Polish if it is separable and completely metrizable. The topology given by Theorem 1.2 will be called the C ∞ topology on M * (n), and the corresponding space is denoted by M ∞ * (n). For each complete connected Riemannian n-manifold M , there is a canonical continuous map ι : M → M ∞ * (n) given by ι(x) = [M, x], which induces a continuous injective map ῑ : Iso(M )\M → M ∞ * (n), where Iso(M ) denotes the isometry group of M . The more explicit notation ι M and ῑM may be also used. The images of the maps ι M form a natural partition of M ∞ * (n), denoted by F * (n). A Riemannian manifold, M , is said to be non-periodic if Iso(M ) = {id M }, and is said to be locally non-periodic if each point x ∈ M has a neighborhood U x such thatLet M * ,np (n) and M * ,lnp (n) be ...

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Section: And We Can Canonically Identify πmentioning

confidence: 99%

mentioning

confidence: 99%

A universal Riemannian foliated space

López¹,

Lijó²,

Candel³

2014

Preprint

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“…A characterization of sequential spaces in terms of convergence of sequences was given by V. Koutnik [15]. We will include the Acta Mathematica Hungarica 123, 2009 SEQUENTIAL CONVERGENCE VIA GALOIS CORRESPONDENCES '% characterization made in [8] because it has a very natural relation with other results in this paper. At the same time, the approach we had in [8] is more related to the (ultra)lter case (see [16]).…”

Section: Sequential Spacesmentioning

confidence: 99%

“…In fact, this is the failure of the rst step of the iteration in (3 ). For details, see condition (3.2) in [8].…”

Section: Sequential Spacesmentioning

confidence: 99%

“…Using appropriate Galois correspondences, we dene pseudosequential spaces (Denition 20) and convergence sequential spaces (Denition 2). In [8], the convergence relations of sequential spaces were characterized using axioms that, at the same time, include the ones of Fréchet/Urysohn and relate to the axioms for lter convergence of topological spaces (see [16]). Following that work, we characterize other sequential classes, as well as their restrictions to topological spaces, which are contained in but are not equal to sequential spaces.…”

Section: Introductionmentioning

confidence: 99%

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Sequential convergence via Galois correspondences

Gutierres

Hofmann

2008

Acta Math Hung

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Topological sequential spaces are the xed points of a Galois correspondence between collections of open sets and sequential convergence structures. The same procedure can be followed replacing open sets by other topological concepts, such as closure operators or (ultra)lter convergences. The xed points of these other Galois correspondences are not topological spaces in general, but they can be embedded into the larger topological classes of pretopological, pseudotopological and convergence spaces.In this paper, we characterize the sequential convergences which are xed points of these correspondences as well as their restrictions to topological spaces.

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