On smooth dendroids

Charatonik, Janusz J.; Eberhart, Carl

doi:10.4064/fm-67-3-297-322

Cited by 52 publications

(30 citation statements)

References 9 publications

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“…In this section we investigate various types of maps on weakly smooth continua and the extent to which they preserve weak smoothness. Compare these results with those of Gordh [8] and Charatonik and Eberhart [3].…”

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confidence: 56%

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Weakly Smooth Continua

Lum¹

1975

Transactions of the American Mathematical Society

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ABSTRACT. We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua.In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.

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supporting

confidence: 56%

“…A dendroid X is smooth at the point p if whenever xn is a net in X the condition lim" xn = x implies Limn \p, xn] = \p, x], where \p, x] denotes the unique subcontinuum of X irreducible between p and x. Smooth dendroids were investigated by Charatonik and Eberhart [3]. The nonmetric analog of smooth dendroids, generalized trees, was studied by Ward [20].…”

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confidence: 99%

Weakly Smooth Continua

Lum¹

1975

Transactions of the American Mathematical Society

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“…It is easy to see that (4) In the remainder we give another method for showing that G({p), X) « ß which will, for example, show that if X is the Cantor fan (cone over the Cantor set), then G({p), X) äs ß where p is the vertex of the cone. The criterion which applies here is an immediate consequence of (1) and (2).…”

Section: Intervals Of Continua Which Are Hilbert Cubes Carl Eberhartmentioning

confidence: 99%

Intervals of continua which are Hilbert cubes

Eberhart¹

1978

Proc. Amer. Math. Soc.

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Abstract. If P is a subcontinuum of a metric continuum X, then by the interval of continua G(P, X) we mean the space of all subcontinua of X which contain P (with the Hausdorff metric). We show that &ÍP, X) is often homeomorphic with the Hilbert cube.In what follows X is a metric continuum and 6(X) is the space of all subcontinua of X with the Hausdorff metric. If F G Q(X), then by the interval of continua G(P, X) we mean the subspace of Q(X) consisting of all A E &(X) which contain P. In [3], Curtis and Schori have shown that if X is locally connected and X \ P is a nonvoid set containing no arc with interior, then the interval Q(P, X) is homeomorphic with the Hilbert cube Q. In this note we will use a recent characterization of ß due to Torunczyk [8] to show that G(P, X) is often homeomorphic with ß without assuming X is locally connected.In order to state Torunczyk's theorem we need his definition of a Z-map. A closed subset A of X is called a Z-set in X provided the identity map on X, lx, can be approximated by maps/: X -> X such that/(X) n A = 0. A map /: X -» X is a Z-map if f(X) is a Z-set in X. The following remarkable result is a special case of Theorem 1 in Torunczyk's paper.

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“…The notion of smoothness was first studied by Charatonik and Eberhart [2] for dendroids and was later generalized in [7] for continua hereditarily unicoherent at a point. In [9] Mackowiak generalized smoothness to metric continua.…”

Section: But Then Cl(m -H) Is Connected and Z E M -H E Cl(m -H)mentioning

confidence: 99%

Monotone Decompositions of Iuc Continua

Collins¹

1984

Transactions of the American Mathematical Society

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Abstract. For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which " improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum M has property IUC provided each proper subcontinuum of M with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.0. Introduction. In [5] FitzGerald and Swingle, by use of set functions, described a monotone upper semicontinuous decomposition 6Da of a compact Hausdorff continuum M such that sHa is the core decomposition of M with respect to having an aposyndetic quotient space. A spectrum of decompositions which " fills in" between M and M/6i)a has been developed for hereditarily unicoherent metric continua by such authors as J. J. Charatonik [1], E. J. Vought [8, 12] and G. R. Gordh [6,8].The main purpose of this paper is to extend the spectrum of decompositions to the class of IUC continua. A compact metric continuum has property IUC provided each proper subcontinuum with interior is unicoherent [3]. If the continuum has property IUC hereditarily then the continuum has property HIUC. Note that the class of IUC continua includes all atriodic continua and hereditarily unicoherent continua.Throughout this paper M will denote a compact metric continuum. An upper semicontinuous (u.s.c.) decomposition ty of M is core with respect to some property P provided tf) is the unique minimal decomposition with respect to which ty has property P. A set function TV is expansive provided that if A and B are subsets of M then A E N(A) and N(A) E N(B) whenever A E B. The subset A is TV-closed provided A = N(A). In [5, Theorem 2.5, p. 37] FitzGerald and Swingle prove that if TV is any expansive set function on M then there exists a core decomposition § of M with respect to the property: § is u.s.c with TV-closed elements. They also note that if TV is monotone then § is monotone.Let <î> be a u.s.c. decomposition of M and % E 6Í). We denote by %* the subset of M consisting of the sum of the elements of %. If B is a subset of M/ty then B~x will

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On smooth dendroids

Cited by 52 publications

References 9 publications

Weakly Smooth Continua

Weakly Smooth Continua

Intervals of continua which are Hilbert cubes

Monotone Decompositions of Iuc Continua

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