The pseudo-arc

Lewis, Wayne

doi:10.1090/conm/117/1112808

Cited by 40 publications

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“…The following construction was suggested by D. Burago; it is based on two ideas: 1) the construction in [5, 3.1], and 2) the construction of a pseudoarc given in [9] (see also the survey [11] and the references therein). In fact, in the first part of the construction, f −1 (0) will be homeomorphic to a pseudoarc, and in the second part f −1 (0) will be homeomorphic to a product of pseudoarcs.…”

Section: Example 42mentioning

confidence: 99%

On intrinsic isometries to Euclidean space

Petrunin

2011

St. Petersburg Math. J.

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Abstract. Compact metric spaces that admit intrinsic isometries to the Euclidean d-space are considered. Roughly, the main result states that the class of such spaces coincides with the class of inverse limits of Euclidean d-polyhedra. §1. IntroductionThe intrinsic isometries are defined in §2; this is a variation of the notion of a path isometry, i.e., a map that preserves the lengths of curves. Any intrinsic isometry is a path isometry; the converse is not true in general.The following statement is a reason for me to prefer intrinsic isometry. Starting Proposition 1.1. Let X be a compact metric space that admits an intrinsic isometry to the d-dimensional Euclidean space (further denoted by E d ). Then dim X ≤ d, where dim denotes the Lebesgue covering dimension.This statement is proved in §3. A similar statement for path isometry fails; see Example 4.2. Furthermore, also the Hausdorff dimension cannot be bounded. For example, the R-tree admits an intrinsic isometry to R and it contains compact subsets of arbitrarily large Hausdorff dimension.Here are some known results on length spaces that admit intrinsic isometry to E d . Theorem 1.2. Let R be a d-dimensional Riemannian space and f : R → E d a short map.1 Then, given ε > 0, there is an intrinsic isometry ı :for any x ∈ R. In particular, any Riemannian d-space admits an intrinsic isometry toFor path isometries, this theorem was proved in [7, 2.4.11], and the same proof works for intrinsic isometries. Applying this theorem, one can show that any limit of an increasing sequence of Riemannian metrics on a fixed d-dimensional manifold admits an intrinsic isometry to E d . (The proof is similar to the "if"-part of the main theorem.) In particular, any sub-Riemannian metric on a d-dimensional manifold admits an intrinsic isometry to E d .

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Section: Example 42mentioning

confidence: 99%

On intrinsic isometries to Euclidean space

Petrunin

2011

St. Petersburg Math. J.

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“…REMARK. In [5], a stronger condition (2 ) meshC i < 1/ i and meshD i < 1/ i is required instead of the condition (2) above. The proof of the theorem easily shows that the same conclusion holds under the condition (2) above.…”

Section: D(m)} Which Follows F and P ∈ D(0)mentioning

confidence: 99%

“…[5] is an excellent survey article on the space, from which we quote all results below. First observe that dim P = 1 by the above condition (a) and the connectedness of P. …”

Section: Introduction Main Theorem and Preliminaries For A Locally mentioning

confidence: 99%

On a Conjecture of Wood

Kawamura

2005

Glasgow Math. J.

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Abstract. We show that there exists a locally compact separable metrizable space L such that C 0 (L), the Banach space of all continuous complex-valued functions vanishing at infinity with the supremum norm, is almost transitive. Due to a result of Greim and Rajagopalan [3], this implies the existence of a locally compact Hausdorff spaceL such that C 0 (L) is transitive, disproving a conjecture of Wood [9]. We totally owe our construction to a topological characterization due to Sánches [8].2000 Mathematics Subject Classification. 54C35, 46B04, 54G99.1. Introduction, main theorem and preliminaries. For a locally compact Hausdorff space L, C 0 (L) denotes the Banach space of all continuous complex-valued functions on L vanishing at infinity, equipped with the supremum norm. A Banach space X is said to be transitive (resp. almost transitive) if the isometry group G(X) acts transitively on the unit sphere S(X) = {x ∈ X| x = 1} (resp. the orbit G(X) · x is dense in S(X) for each x ∈ S(X)). In [9], Wood conjectured that C 0 (L) is not transitive for any locally compact Hausdorff space L unless L is a singleton. In [3], the conjecture was verified for the Banach space C 0 (L : R) of all real-valued continuous functions on L vanishing at infinity. Greim and Rajagopalan proved in [3] that the existence of a locally compact Hausdorff space L with C 0 (L) being almost transitive implies the existence of a locally compact Hausdorff spaceL such that C 0 (L) is transitive. In [7], the verification of the conjecture was reduced to the case in which L has the metrizable onepoint compactification αL. Furthermore, Sánches in [8] gave an explicit topological characterization of the space L with the almost transitive C 0 (L). Theorem 2 and 3 of [8] are restated below. Following [8], we say that a locally compact Hausdorff space L is an Wood space (resp. an almost Wood space) if C 0 (L) is transitive (resp. almost transitive).

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“…In [8] Lewis asked whether curves admitting continuous decomposition into pseudo-arcs with an arc as the quotient space are topologically unique (Question 1), and more specifically, whether the elements of such a decomposition must be terminal (Question 2). The above examples show that the answer to both these question is no.…”

Section: Theorem 8 There Exist a Plane S-arc Of Pseudo-arcs And A Plmentioning

confidence: 99%

“…Further, as a consequence of Bing's [3] characterization of the pseudo-arc, it turned out to be topologically equivalent to the Knaster curve. Moise's result was a starting point to intensive research on this very special continuum by a number of authors (see the survey paper [8]). In this paper we refer to some investigations made by Knaster before Moise's construction.…”

mentioning

confidence: 98%

A continuous circle of pseudo-arcs filling up the annulus

Prajs

1999

Trans. Amer. Math. Soc.

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Abstract. We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continuous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, "a selectible circle of pseudo-arcs", and answers some questions of Lewis.In 1922 the famous construction of an hereditarily indecomposable plane continuum was presented [6] by B. Knaster. Twenty-five years later Moise [11] constructed an hereditarily equivalent and hereditarily indecomposable plane continuum, and called it a pseudo-arc. Further, as a consequence of Bing's [3] characterization of the pseudo-arc, it turned out to be topologically equivalent to the Knaster curve. Moise's result was a starting point to intensive research on this very special continuum by a number of authors (see the survey paper [8]). In this paper we refer to some investigations made by Knaster before Moise's construction. Among the results obtained by Knaster during World War II one can find the following announcement, originally presented in Kiev in 1940:There exists a real-valued, monotone mapping from the plane that is not constant on any arc.In the construction Knaster's hereditarily indecomposable continua were exploited. Actually, Knaster's result can be reformulated in the following stronger version (see [10], p. 225):There exists a real valued, monotone mapping from the plane such that all pointinverses are hereditarily indecomposable.Unfortunately, Knaster's notes concerning this result were burned during the war, Knaster had never written down the result again, and even his closest exstudents do not know his original idea of construction.A result similar to that announced by Knaster (with higher dimensional analogues) was proved by Brown [5] in 1958. In fact, a continuous decomposition into hereditarily indecomposable continua of each Euclidean n-space with one point deleted was constructed, such that the real line was the quotient space.Having no confidence that we follow Knaster's idea, in this paper we construct an example of an open mapping as in the announcement. Then we use it to obtain a continuous decomposition of the plane band (annulus) into pseudo-arcs, such that

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The pseudo-arc

Cited by 40 publications

References 0 publications

On intrinsic isometries to Euclidean space

On intrinsic isometries to Euclidean space

On a Conjecture of Wood

A continuous circle of pseudo-arcs filling up the annulus

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