Symmetries of two-point sets

Chad, Ben; Suabedissen, Rolf

doi:10.1016/j.topol.2008.02.009

Cited by 3 publications

(4 citation statements)

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“…Note that as θ → θ 0 ± π/2, then d(θ ) → 0 and r → ∞. Hence by continuity it must be the case that (q z − q y )(θ 0 ± π/2) + θ z − θ y ≡ 0 (mod π), (1) for otherwise |t x (θ)| → ∞ as θ → θ 0 ± π/2 and t x dominates d, which is a contradiction. By applying L'Hôpital's Rule, we see that the limit of t x (θ) as θ → θ 0 ± π/2 exists and is finite.…”

Section: The Existence Of Homogeneous Two-point Setsmentioning

confidence: 98%

“…It is shown in [1] that any subgroup of S 1 of cardinality less than c may be realised as the isometry group of a two-point set, and that there exist subgroups of S 1 of size c which are the isometry group of a two-point set. Larman [2] proved that a two-point set cannot contain an arc, and so a two-point set cannot have isometry group S 1 .…”

Section: Subgroups Of S 1 and The Isometry Group Of A Two-point Setmentioning

confidence: 99%

“…(A French translation is available in [4]. ) Chad and Suabedissen [1] have asked "What are the symmetries of a two-point set?". They showed that two-point sets may be rigid and that the isometry group of a two-point set consists of rotations only.…”

Section: Introductionmentioning

confidence: 99%

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On affine groups admitting invariant two-point sets

Chad¹,

Suabedissen²

2009

Topology and its Applications

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Section: The Existence Of Homogeneous Two-point Setsmentioning

confidence: 98%

Section: Subgroups Of S 1 and The Isometry Group Of A Two-point Setmentioning

confidence: 99%

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On affine groups admitting invariant two-point sets

Chad¹,

Suabedissen²

2009

Topology and its Applications

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“…The second motivation for our work is a desire to better understand the structure of two-point sets and to follow a line of research started by Chad and Suabedissen [4,5]. These papers have studied autohomeomorphims of two-point sets, and their main results include the facts that two-point sets may be chosen to be rigid or homogeneous or to have isometry group isomorphic to any subgroup of S 1 of cardinality less than c. These results examine a two-point set by looking for similarity within itself; our results will examine a two-point set by looking for similarity with other distinct types of geometric objects.…”

Section: Introductionmentioning

confidence: 99%

Homeomorphisms of two-point sets

Chad

Good

2011

Proc. Amer. Math. Soc.

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Abstract. Given a cardinal κ ≤ c, a subset of the plane is said to be a κ-point set if and only if it meets every line in precisely κ many points. In response to a question of Cobb, we show that for all 2 ≤ κ, λ < c there exists a κ-point set which is homeomorphic to a λ-point set, and further, we also show that it is consistent with ZFC that for all 2 ≤ κ < c, there exists a κ-point set X such that for all 2 ≤ λ < c, X is homeomorphic to a λ-point set. On the other hand, we prove that it is consistent with ZFC that for all 2 ≤ κ, λ < c, there exists a κ-point set, such that for all homeomorphisms f : R 2 → R 2 , if f (X) is a λ-point set, then λ = κ.

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Set-theoretic constructions of two-point sets

Chad

Knight²,

Suabedissen³

2009

Fund. Math.

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Abstract. A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice.1. Introduction. Given a cardinal κ, a subset of the plane is said to be a κ-point set if and only if it meets every line in exactly κ many points and is said to be a partial κ-point set if and only if it meets every line in at most κ many points. We are particularly interested in the case that κ = 2, and we refer to such sets as two-point sets.The existence of two-point sets was first shown by Mazurkiewicz [8]. (A French translation is given in [9].) A two-point set is classically constructed via transfinite recursion, making use of an arbitrary well-ordering of the collection of all lines in the plane. We will demonstrate two constructions of two-point sets which result by varying the use of this well-ordering.The first construction shows that working in ZFC+CH, we can construct a two-point set which is contained in the union of a countable collection of concentric circles of unbounded radius. In general, classical constructions of two-point sets can be thought of as starting with a c-point set and then refining it to a two-point set, and so this construction is of interest because we obtain a two-point set by refining an ℵ 0 -point set. We achieve this development by being careful about the well-ordering of the collection of all lines that we choose to help us in our construction.The second construction shows that in ZF, if we assume that the real line satisfies a certain number of properties, we can construct a two-point set without explicitly invoking the Axiom of Choice. It is a question of Erdős,

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Symmetries of two-point sets

Cited by 3 publications

References 5 publications

On affine groups admitting invariant two-point sets

On affine groups admitting invariant two-point sets

Homeomorphisms of two-point sets

Set-theoretic constructions of two-point sets

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