Confluent and related mappings

Read, D. R.

doi:10.4064/cm-29-2-233-239

Cited by 22 publications

(6 citation statements)

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“…It is known that chainable continua satisfy conditions (i), (ii), (iii) and (v) (see [14], [21] and (2.3)), and that hereditarily indecomposable continua satisfy conditions (i)-(v) (see [14], [5], [15] and (2.4)). However, there exist examples of continua which do not have to satisfy any of the above conditions.…”

Section: Applications To Circle-like Continua and Smooth Dendroidsmentioning

confidence: 99%

Some Properties of Hyperspaces with Applications to Continua Theory

Grispolakis¹,

Nadler²,

Tymchatyn³

1979

Can. j. math.

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In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainableIn [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

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Section: Applications To Circle-like Continua and Smooth Dendroidsmentioning

confidence: 99%

Some Properties of Hyperspaces with Applications to Continua Theory

Grispolakis¹,

Nadler²,

Tymchatyn³

1979

Can. j. math.

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“…Lelek [8] introduced the notion of Class W continua in 1972, several people have tried to determine which continua are contained in Class W and have looked for different characterizations of Class W continua. H. Cook [1] proved that all hereditarily indecomposable continua are in Class W. D. R. Read [10] showed that each chainable continuum is in Class W. G. A. Feuerbacher [2] proved that each nonplanar circle-like continuum is in Class W. B. Hughes [3] proved that Class W contains all continua which have the covering property. J. Grispolakis and E. D. Tymchatyn proved that Class W contains atriodic tree-like continua [6], compactifications of the half line [0, co) with remainder a continuum in Class W [3], and irreducible continua of type A that have each tranche as a tranche of cohesion and each nondegenerate tranche in Class W [3].…”

Section: Introductionmentioning

confidence: 99%

A characterization of absolutely 𝐶*-smooth continua

Proctor¹

1984

Proc. Amer. Math. Soc.

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A continuum X X is proven to be absolutely C ∗ {C^*} -smooth if and only if each compactification Y Y of the half line [ 0 , ∞ ) [0,\infty ) with remainder X X has the property that the space of all subcontinua of Y Y is a compactification of the space of all subcontinua of [ 0 , ∞ ) [0,\infty ) .

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“…Applications of such results are in fixed point theory and the structure of continua. For example: In 2.11 of [12] we showed that a certain induced map is universal; then we obtained Segal's fixed point theorem (Theorem 3 of [15]) and Read's theorem about maps onto arc-like continua (Theorem 4 of [14]) as immediate corollaries. We remark that universal maps are used in a disguised form in dimension theory-the AlexanderHopf maps onto n-cells are precisely the universal maps onto n-cells (1.3 of [12]).…”

Section: Introductionmentioning

confidence: 99%

Induced universal maps and some hyperspaces with the fixed point property

Nadler¹

1987

Proc. Amer. Math. Soc.

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ABSTRACT. For a (metric) continuum Z, let 2Z (resp., C(Z)) denote the space of all nonempty compacta (resp., continua) in Z with the HausdorfT metric. We prove: (1) If / is a monotone map of a continuum X onto a Peano continuum Y, then, for any maps g: 2X -> 2Y and h: C(X) -► C(V), there is A 6 2X and B € C(X) such that f(A) = g(A) and /(B) = h{B). We use (1) to prove: (2) If X is an inverse limit of dendrites with quasi-monotone bonding maps, then 2X and C(X) have the fixed point property. Thus, we have a proof that for certain indecomposable continua X, 2X has the fixed point property.

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Confluent and related mappings

Cited by 22 publications

References 0 publications

Some Properties of Hyperspaces with Applications to Continua Theory

Some Properties of Hyperspaces with Applications to Continua Theory

A characterization of absolutely 𝐶*-smooth continua

Induced universal maps and some hyperspaces with the fixed point property

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