On weakly chainable continua

Lelek, A.

doi:10.4064/fm-51-3-271-282

Cited by 33 publications

(25 citation statements)

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“…First note that M arises in a condensation process from a ray spiralling on a circle. Letting A denote the latter continuum it is not difficult to see that A is the continuous image of M. Therefore ψ cannot be mapped onto M; otherwise, we could map ψ onto A and this is impossible (see [17,Example 2]). Furthermore, Whyburn shows that (1) every subcontinuum of M contains a continuum which is homeomorphic to M and (2) if any subcontinuum N of M contains a point of a subcontinuum W of M which is homeomorphic to M, then N contains W or W contains N. We use C(X) to denote the space of all nonvacuous subcontinua of a continuum X metrized by the Hausdorff metric (see [14]).…”

Section: Quasi Dimension Type Il Types In 1-dimensional Spaces Jack mentioning

confidence: 99%

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Quasi dimension type. II. Types in 1-dimensional spaces

Segal¹

1968

Pacific J. Math.

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Section: Quasi Dimension Type Il Types In 1-dimensional Spaces Jack mentioning

confidence: 99%

“…However, this does not exhaust the class of snake-like continua. Bing [3], Lelek [17], and others have shown that ψ can be mapped onto any snake-like continuum. We sharpen this result by proving that, for any ε > 0, ψ can be ε-mapped onto any snake-like continuum.…”

Section: Quasi Dimension Type Il Types In 1-dimensional Spaces Jack mentioning

confidence: 99%

Quasi dimension type. II. Types in 1-dimensional spaces

Segal¹

1968

Pacific J. Math.

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“…Does every periodic homeomorphism h of the pseudo-arc have an nth-root for each positive integer n? If n > 1 and kn = h, where h has a nondegenerate fixed point set, can be the fixed point set of k always be made smaller than that of hi A continuum is weakly chainable [7,8] if it is the continuous image of a chainable continuum, in particular the image of a pseudo-arc. Question 7.…”

Section: Pth Roots Of Homeomorphismsmentioning

confidence: 99%

Most Maps of the Pseudo-Arc are Homeomorphisms

Lewis¹

1984

Proceedings of the American Mathematical Society

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Abstract.We prove the following results.(1) If M(P) is the space of maps of the pseudo-arc into itself with the sup metric, then the subset H(P) of maps of the pseudo-arc into itself which are homeomorphisms onto their images is a dense Gs in M(P). (2) Every homeomorphism of the pseudo-arc onto itself is a product of e-homeomorphisms.(3) There exists a nonidentity homeomorphism of the pseudo-arc with an infinite sequence of pth roots. (4) Every map between chainable continua can be lifted to a homeomorphism of pseudo-arcs.We will investigate certain properties of maps and homeomorphisms of the pseudo-arc and other chainable continua. The pseudo-arc is characterized [2] as a nondegenerate, hereditarily indecomposable, chainable continuum. In most cases we will be defining maps or investigating their behavior in terms of functions between defining sequences of chains and patterns. For definitions of chain, link, pattern, amalgamation, and crooked, and for basic properties of these, the reader is referred to [1, 2, 9, and 12]. Unless otherwise specified all maps are surjections.Some specific facts we will make use of are the following.Lemma 1 [2]. If X is a nondegenerate chainable continuum such that for each chain C covering X and e > 0 there is a chain D of mesh less than e which covers X and is crooked in C, then X is a pseudo-arc.

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“….. . In [16] it is proven that a continuum is weakly chainable if and only if it is a continuous image of the pseudo arc (cf. [17 and 7]).…”

Section: Introductionmentioning

confidence: 99%