Decompositions of homogeneous continua

Rogers, James

doi:10.2140/pjm.1982.99.137

Cited by 23 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, use [4, Theorem 2]: if a nondegenerate homogeneous continuum X contains an arc and small subcontinua of X are atriodic, then X is a solenoid. Theorem 3.1 improves the characterizations of [10, (3.8)], [11, (4.8)], [15] and [16], since nontrivial terminal subcontinua of homogeneous curves are hereditarily indecomposable (see the next section).…”

Section: A Characterization Of Solenoidssupporting

confidence: 56%

The disjoint arcs property for homogeneous curves

Krupski¹

1995

Fund. Math.

View full text Add to dashboard Cite

The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and small subcontinua are not ∞-ods. Introduction.All spaces in the paper are metric separable and all maps are continuous. A space X is homogeneous if for each two points x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. A curve is a one-dimensional continuum. A space X has the disjoint arcs property (DAP) if any two paths in X can be approximated, arbitrarily closely, by disjoint paths. More precisely, X has the DAP if for each ε > 0 and for any two maps f, g : I = [0, 1] → X there exist maps f , g : I → X such that f (I)∩g (I) = ∅ and (f, f ) < ε, (g, g ) < ε, where denotes the sup-norm metric induced by in X. If, in the definition, I is replaced by the ndimensional disk, one gets the disjoint n-disks property (DD n P ). The latter property turned out to be crucial in recognizing n-dimensional Menger space manifolds among LC n−1 -spaces [2]. The DD n P is a kind of a general position property, so it requires enough room-for manifolds this is guaranteed by the sufficiently high dimension, whereas Menger space manifolds have a special structure which is responsible for the DD n P.In [8] it was observed that all homogeneous locally compact locally connected spaces of dimension at least three have the DAP. Dimension two is not friendly to the DAP because no two-dimensional manifold has the property.

show abstract

Section: A Characterization Of Solenoidssupporting

confidence: 56%

The disjoint arcs property for homogeneous curves

Krupski¹

1995

Fund. Math.

View full text Add to dashboard Cite

show abstract

“…In forthcoming papers, the authors will present applications of the material thus developed. Notably, we give: a scheme to categorize homogeneous continua into subclasses defined in terms of filament and ample subcontinua which compares interestingly to those given by Rogers in [22,24]; aposyndetic decompositions of Kelley continua extending Jones' decomposition of homogeneous continua [7]; newly defined decompositions of homogeneous continua related to those given in [7,3,20,21,23]; and applications to questions of aposyndesis, local connectedness, and indecomposability.…”

Section: Questionmentioning

confidence: 99%

“…Interest in homogeneous continua over the decades has opened new areas of mathematical research going far beyond homogeneity. Despite the obvious attention, progress in classifying homogeneous continua could be characterized as slow with sporadic bursts of activity resulting from the introduction of new tools such as the aposyndetic decomposition theorem of Jones [7], its improvements by Rogers [20,21,23], and the Theorem of Effros [4,Theorem 2]. The present case appears to be another such event.…”

mentioning

confidence: 98%

Filament sets and homogeneous continua

Prajs¹,

Whittington²

2007

Topology and its Applications

View full text Add to dashboard Cite

New tools are introduced for the study of homogeneous continua. The subcontinua of a given continuum are classified into three types: filament, non-filament, and ample, with ample being a subcategory of non-filament. The richness of the collection of ample subcontinua of a homogeneous continuum reflects where the space lies in the gradation from being locally connected at one extreme to indecomposable at another. Applications are given to the general theory of homogeneous continua and their hyperspaces. This is the introductory paper in a series of articles [18,17,19,16] which develop and apply new tools in continuum theory based on the concepts of filament and ample subcontinua. These techniques are especially suited to studying homogeneous continua as well as a larger, well known, class of spaces, which we refer to as Kelley continua.The topic of homogeneous continua has been a classic since 1920, when Knaster and Kuratowski asked about such continua in the plane [9]. Homogeneous continua form a natural class of spaces, with remarkable examples such as the pseudo-arc, solenoids, and the Menger universal curve, to mention just a few. Interest in homogeneous continua over the decades has opened new areas of mathematical research going far beyond homogeneity. Despite the obvious attention, progress in classifying homogeneous continua could be characterized as slow with sporadic bursts of activity resulting from the introduction of new tools such as the aposyndetic decomposition theorem of Jones [7], its improvements by Rogers [20,21,23], and the Theorem of Effros [4, Theorem 2]. The present case appears to be another such event. The tools which we offer also connect with another, quite different, area of intense interest in topology: the hyperspace of subcontinua of a given continuum.In a continuum X, the subcontinua fall naturally into two mutually exclusive types which we call filament and non-filament. A subcontinuum F is filament if it has a neighborhood in which the component containing F has empty

show abstract

“…After its discovery in 1975 by Ungar [13] it has been used in several papers on homogeneous spaces written after 1975. As the work of Hagopian [3,4], Ungar [13,14], Jones [7], Rogers [11], Lewis [8,9], Phelps [10] and Ancel [1] shows, many old problems have been solved and some old proofs have been simplified by the use of Effros's theorem in the form of microtransitivity.…”

mentioning

confidence: 99%