An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua

Moïse, Edwin E.

doi:10.1090/s0002-9947-1948-0025733-4

Cited by 88 publications

(39 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If M is aposyndetic, it is an arc because it is irreducible between some pair of its points. If M is non-aposyndetic it may lie at the other end of the spectrum (as shown by Moise's example of a pseudo-arc) [9]. But must it?…”

Section: Example 10mentioning

confidence: 99%

“…I would like to point out that one cannot raise the dimension of a Knaster continuum (and keep it a Knaster continuum) by means of Cartesian products for then they become aposyndetic. 9 As I indicated in the introduction, we have here only a beginning of a classification of continua by means of their aposyndetic properties. I have indicated one way, but not a very fruitful way, of refining this classification.…”

Section: Example 10mentioning

confidence: 99%

See 1 more Smart Citation

Concerning aposyndetic and non-aposyndetic continua

Jones¹

1952

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Section: Example 10mentioning

confidence: 99%

Section: Example 10mentioning

confidence: 99%

Concerning aposyndetic and non-aposyndetic continua

Jones¹

1952

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

“…The space was constructed by R. H. Bing [1] and E. E. Moise [6]. It has been playing an important role in continuum theory (in topology) and our result is another simple application of the topology of the pseudo-arc.…”

Section: Introduction Main Theorem and Preliminaries For A Locally mentioning

confidence: 98%

On a Conjecture of Wood

Kawamura

2005

Glasgow Math. J.

View full text Add to dashboard Cite

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).

show abstract

“…For many embeddings, e.g., the "standard" embedding with two accessible composants [7], one can show that every extendable homeomorphism must preserve orientation of the plane. Thus our example depends on choosing the correct embedding of the pseudo-arc.…”

Section: Introductionmentioning

confidence: 99%

An orientation reversing homeomorphism of the plane with invariant pseudo-arc

Bellamy¹,

Lewis²

1992

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.Using the universal cover of the Möbius band, we construct an example as described in the title which answers a question of K. Kuperberg.Krystyna Kuperberg has recently asked whether there exists an orientation reversing homeomorphism of the plane with an invariant pseudo-arc. We provide a positive answer to this question. Her question was apparently motivated by her recent work [4] on fixed points and orientation reversing homeomorphisms.Most embeddings of the pseudo-arc in the plane appear to admit very few homeomorphism which extend to homeomorphisms of the plane. For many embeddings, e.g., the "standard" embedding with two accessible composants [7], one can show that every extendable homeomorphism must preserve orientation of the plane. Thus our example depends on choosing the correct embedding of the pseudo-arc.Since there has been significant recent interest in the dynamics of homeomorphisms of the pseudo-arc and in which homeomorphisms of the pseudo-arc extend to homeomorphisms of the plane, it is worth noting that the example we construct has fairly elementary dynamics. There are two fixed points of the pseudo-arc under our homeomorphism.Every other point of the psuedo-arc is repelled from one of these fixed points and attracted to the other. One can construct examples with more interesting dynamics.We conclude with some observations about prime ends, accessible points, and extendable homeomorphisms. The constructionOur main result is the following Theorem. There exists an orientation reversins homeomorphism of the plane with an invariant pseudo-arc.Our proof is constructive. The construction can be outlined in the following few steps and observations. -

show abstract

An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua

Abstract: , and before .it was submitted for publication, R. H. Bing observed that the continua described also settle the question whether every bounded homogeneous plane continuum is a simple closed curve. He is publishing this result elsewhere.

Cited by 88 publications

References 0 publications

Concerning aposyndetic and non-aposyndetic continua

Concerning aposyndetic and non-aposyndetic continua

On a Conjecture of Wood

An orientation reversing homeomorphism of the plane with invariant pseudo-arc

Contact Info

Product

Resources

About