Monotone decompositions of irreducible Hausdorff continua

“…In Theorem 5 the decomposition g is shown to be unique and minimal with respect to the properties of being monotone, upper semicontinuous and having a semiaposyndetic quotient space. But Gordh [5,Theorem 2.3] has proved that a semiaposyndetic, hereditarily unicoherent continuum is arcwise connected. So it follows from this that & ^ §.…”

Section: Introductionmentioning

confidence: 99%

Monotone decompositions of Hausdorff continua

Vought¹

1976

Proc. Amer. Math. Soc.

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Abstract.A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of M are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum M has a unique, minimal admissible decomposition <£. For hereditarily unicoherent continua & is also the unique, minimal decomposition with respect to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition § is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then & refines § and S refines the unique, minimal decomposition £ of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua).For a compact, Hausdorff continuum M, FitzGerald and Swingle [4, p. 37] have obtained a unique monotone, upper semicontinuous decomposition £ whose quotient space (M, £) is semilocally connected and which is minimal with respect to these properties. A second description of this decomposition using collections of closed subsets which separate M is given by them in [4, p. 49] and by McAuley in [9, p. 2]. For a compact, metric continuum M, Charatonik has defined a decomposition to be admissible if it is monotone, upper semicontinuous and the layers of the irreducible subcontinua of M are contained in the elements of the decomposition [2, pp. 115-116]. He then proves in the same paper that the quotient space of an admissible decomposition of a continuum M is hereditarily arcwise connected and that every continuum has a unique, minimal admissible decomposition &. One purpose of this paper is to extend Charatonik's results to compact, Hausdorff continua.A second purpose pertains to the class of hereditarily unicoherent, Hausdorff continua. It is easily seen with simple examples that in general the two decompositions mentioned above are not comparable, i.e., neither refines the other. However, if the continuum M is hereditarily unicoherent then & refines £ (61 g £) since the semilocally connected quotient space (M, £) must then be arcwise connected. In this paper a third decomposition S is constructed which is the unique minimal decomposition with respect to being monotone, upper semicontinuous and having a semiaposyndetic quotient space. The construction involves a collection of closed subsets of M that separate M and satisfies

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“…Clearly every subcontinuum of a tree X is a tree, and consequently X is hereditarily locally connected. It follows immediately from Theorem 4.1(3) of [4] that a tree is a generalized arc if and only if it is atriodic. It is known that the limit of a monotone inverse system of trees is a tree (see the proof of Theorem 4.2 in [4]); and that the limit of a monotone inverse system of generalized arcs is a generalized arc (Lemma 4.7 of [1], or [8]).…”

mentioning

confidence: 98%

“…It follows immediately from Theorem 4.1(3) of [4] that a tree is a generalized arc if and only if it is atriodic. It is known that the limit of a monotone inverse system of trees is a tree (see the proof of Theorem 4.2 in [4]); and that the limit of a monotone inverse system of generalized arcs is a generalized arc (Lemma 4.7 of [1], or [8]). We shall obtain the same conclusions for σ-directed inverse systems of trees and generalized arcs without any assumptions about the bonding maps.…”

mentioning

confidence: 98%

Characterizing local connectedness in inverse limits

Gordh¹,

Mardešić²

1975

Pacific J. Math.

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Let X denote the limit of an inverse system X -{X a p aa >; A} of locally connected Hausdorff continua. The main purpose of this paper is to define a notion of local connectedness for inverse systems, and to prove that if X is locally connected, then so is the limit X. If the bonding maps p aa > are surjections, then X is locally connected if and only if X is. The following corollaries are obtained. (1) If X is σ-directed and surjective, then X is locally connected. (2) If X is well-ordered, surjective, and weight (X«) ^ λ for each a in A, then either weight (X) ^ λ, or X is locally connected. (3) If X is σ -directed and the factor spaces X α are trees (generalized arcs), then X is a tree (generalized arc). (4) If X is well-ordered and the factor spaces X« are dendrites (arcs), then either X is metrizable, or X is a tree (generalized arc).1. Introduction. By a continuum we mean a compact connected Hausdorff space. Let X denote the limit of an inverse system X = {X α ; p αα A} where the factor spaces X a are locally connected continua, and A is an arbitrary directed set. It is well-known that every continuum X can be obtained as the limit of such a system where the factor spaces are polyhedra (see Theorem 10.1, p. 284, [2]). Hence local connectedness of the factor spaces X a does not imply local connectedness of the limit X. It is the main purpose of this paper to introduce a notion of local connectedness for inverse systems, and to prove that for such systems X the limit space X is locally connected (see Theorem 1). The converse holds if X is a surjective system, i.e., if the bonding maps p aa , are surjections. An immediate corollary is the known result that if X is a monotone inverse system, then X is locally connected [1].In §3 the main theorem is applied to well-ordered and σ-directed inverse systems, i.e., systems in which every countable subset of the index set is bounded above. The following somewhat surprising results are obtained. (1) If the inverse system X is ςr-directed and surjective, then the limit X is locally connected. (2) If X is well-ordered, surjective, and weight (X tt ) = λ for each a in A, then weight (X) Si λ or X is locally connected.Section 4 contains similar results about well-ordered and σ-directed inverse systems of trees (i.e., locally connected, hereditarily unicoherent continua [9]) and generalized arcs (i.e., ordered continua).

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Monotone decompositions of irreducible Hausdorff continua

Cited by 14 publications

References 8 publications

Monotone decompositions into trees of Hausdorff continua irreducible about a finite subset

Monotone decompositions into trees of Hausdorff continua irreducible about a finite subset

Monotone decompositions of Hausdorff continua

Characterizing local connectedness in inverse limits

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