Unicoherent Continua Research Articles

Hyperspaces of continua with connected boundaries in π-Euclidean Peano continua

Let X be a nondegenerate Peano unicoherent continuum. The family CB(X) of proper subcontinua of X with connected boundaries is a Gδ-subset of the hyperspace C(X) of all subcontinua of X. If every nonempty open subset of X contains an open subset homeomorphic to Rn (such space is called π-n-Euclidean) and 2≤n<∞, then C(X)∖CB(X) is recognized as an Fσ-absorber in C(X); if additionally, no one-dimensional subset separates X, then the family of all members of CB(X) which separate X is a D2(Fσ)-absorber in C(X), where D2(Fσ) denotes the small Borel class of differences of two σ-compacta.

Mappings of terminal continua

The concept of a terminal continuum introduced in 1973 by G. R. Gordh Jr., for hereditarily unicoherent continua is extended to arbitrary continua. Mapping properties of these two concepts are investigated. Especially the invariance of terminality under mappings satisfying some special conditions is studied. In particular, we conclude that the invariance holds for atomic mappings.

Finite unions of shore sets

Adendroid is an arcwise connected hereditarily unicoherent continuum. Ashore set in a dendroidX is a subsetA ofX such that, for each e>0, there exists a subdendroidB ofX such that the Hausdorff distance fromB toX is less then e andB∩A=θ.

Irreducibility of inverse limits on intervals

A procedure for obtaining points of irreducibility for an inverse limit on intervals is developed. In connection with this, the following are included. A se- miatriodic continuum is defined to be a continuum that contains no triod with interior. Characterizations of semiatriodic and unicoherent continua are given, as well as necessary and sufficient conditions for a subcontinuum of a semiatriodic and unicoherent continuum M to lie within the interior of a proper subcontinuum ofM.

Hereditarily weakly confluent maps and a characterization of Class [formula omitted

First it is proved that the image of any hereditarily unicoherent continuum under a hereditarily weakly confluent map is also hereditarily unicoherent. This answers a question raised in 1979 by T. Maćkowiak. Second, a characterization is established for Class HW, i.e., for the class of all continua Y such that each mapping from a continuum onto Y is hereditarily weakly confluent. That characterization is used to prove J.F. Davis' theorem that atriodic, acyclic curves are in Class HW.

Density of arc components in unicoherent continua

Research of C.L. Hagopian and of J. Krasinkiewicz and P. Minc concerning density of arc components in planar λ-dendroids is generalized to apply to semi-hereditarily unicoherent, ω-connected plane continua. Examples are given showing limits to possible further generalizations and answering a question, of Hagopian and the authors, about conditions that assure that an almost arcwise connected continuum is an almost Peano continuum.

Outlet Points and Homogeneous Continua

(1) A proof is presented for Bing’s conjecture that homogeneous, treelike continua are hereditarily indecomposable. As a consequence, each homogeneous curve admits the continuous decomposition into the maximal terminal, homeomorphic, homogeneous, hereditarily indecomposable, treelike subcontinua. (2) A homogeneous, hereditarily unicoherent continuum contains either an arc or arbitrarily small, nondegenerate, indecomposable subcontinua. (3) A treelike continuum with property $K$ which is homogeneous with respect to confluent light mappings contains no two nondegenerate subcontinua with the one-point intersection.

Outlet points and homogeneous continua

(1) A proof is presented for Bing’s conjecture that homogeneous, treelike continua are hereditarily indecomposable. As a consequence, each homogeneous curve admits the continuous decomposition into the maximal terminal, homeomorphic, homogeneous, hereditarily indecomposable, treelike subcontinua. (2) A homogeneous, hereditarily unicoherent continuum contains either an arc or arbitrarily small, nondegenerate, indecomposable subcontinua. (3) A treelike continuum with property K K which is homogeneous with respect to confluent light mappings contains no two nondegenerate subcontinua with the one-point intersection.

Theorems on tangents to subvarieties of complex tori

E. H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). S. B. Nadler, Jr., "Examples of fixed point free maps from cells onto larger cells and spheres," Rocky Mountain J. Math., ii, No. 2, 319-325 (1981). H. C. Miller, "On unicoherent continua," Trans. Am. Math. Soc., 69, No. i, 179-194 (1950). S. A. Bogatyi, "Periodic points of a mapping of a system of intervals of the line," V. Tiraspol Symposium on General Topology and Applications, 26-28, Kishinev: Shtiintsa (1985). A. N. Sharkovskii, "Coexistence of cycles of a continuous mapping of the line into itself," Ukr. Mat. Zh., 16, No. i, 61-71 (1964). B. Knaster, C. Kuratowski, and S. Mazurkiewicz, "Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe," Fund, Math., 14, 132-137 (1929).

A characterization of semispan of continua

The main results of this paper are characterizations of semispan ε , ε ⩾ 0 \varepsilon ,\varepsilon \geqslant 0 , for those metric spaces which are atriodic hereditarily unicoherent continua. The results follow from Theorem 1 which gives conditions under which the union of two continua, each of semispan less than or equal to ε \varepsilon , has semispan less than or equal to ε \varepsilon .

Almost arcwise connectivity in unicoherent continua

K.R. Kellum has proved that a continuum is an almost continuous image of the interval [0, 1] if and only if it is an almost Peano continuum. Hence, a continuum is an almost continuous image of [0, 1] if it has a dense arc component. Our principal result is that any almost arcwise connected, semi-hereditarily unicoherent, metric continuum with only countably many arc components has a dense arc component. An example is given to show that this is not true for unicoherent continua in general. It is also shown that any semi-hereditarily unicoherent continuum with only countably many arc components has at most one dense arc component, and if it has a dense arc component, then every other arc component is nowhere dense. This generalizes results of Fugate and Mohler for λ-dendroids.

Extension theorems for some classes of continua

It is proved that every mapping from a proper subcontinuum of a hereditarily unicoherent continuum X onto the Knaster's indecomposable continuum (onto a cone over a zerodimensional compact metric set) can be extended to a mapping defined on X. Similarly, every mapping from a proper subcontinuum of a hereditarily indecomposable continuum onto a pseudoarc can be extended to a mapping defined on the whole space. Both of the above results are generalizations of the author's earlier results to the nonmetric case. As a consequence it is obtained that a pseudoarc is continuously n-homogeneous.

Disk-like products of 𝜆 connected continua. II

R. H. Bing [3] proved that every atriodic, hereditarily decomposable, hereditarily unicoherent continuum is arc-like. Using this theorem, the author [5] showed that λ \lambda connected continua X X and Y Y are arc-like when the topological product X × Y X \times Y is disk-like. In this paper we consider products that have a more general mapping property. Suppose that X X and Y Y are λ \lambda connected continua and that for each ε &gt; 0 \varepsilon &gt; 0 , there exists an ε \varepsilon -map of X × Y X \times Y into the plane. Then X X is either arc-like or circle-like. Furthermore, if X X is circle-like, then Y Y is arc-like. Hence X × Y X \times Y is either disk-like or annulus-like.

Concerning closed quasi-orders on hereditarily unicoherent continua

Concerning closed quasi-orders on hereditarily unicoherent continua

Can a 2-coherent Peano continuum separate 𝐸³?

The fact that there are unicoherent continua which separate E 2 {E^2} is well known, e.g., a circle with a spiral converging onto it is such a continuum. In this paper we extend this pathology by describing a Peano continuum which separates E 3 {E^3} and has the property that however it is written as the union of two unicoherent Peano continua, their intersection is unicoherent.

A partition theorem for collections of universal subcontinua

This paper is concerned with conditions under which a collection of universal subcontinua may be partitioned into finitely many subcollections, each of which has the finite intersection property. Throughout X will denote a Hausdorff topological space. A continuum A c X will be called a universal subcontinuum of (USC) X if A N B is a continuum for each continuum B c X. (Ordinarily it is assumed that X is a continuum, but this assumption is unnecessary for our purposes.) WALLACE, [3], studied USC under the title semi-chains. We are concerned with the following topological-combinatorial problem. If ~ is a collection of universal subcontinua, if n_->2 is a positive integer, and if of each n distinct elements of ct at least two have a non-empty intersection, does there exist an integer m <_- n -- 1 and m subcollections cq ..... at,, of ~, each of which has the finite intersection property, such that ct = ~ U -.- U ~t,,? We have not been able to answer this question in general, but in this paper we will answer it affirmatively for n = 3. The importance of the present problem lies in the fact that if ct is a collection of USC which has the finite intersection property then the intersection over ~ is a non-empty USC. The theorem presented here is of the type which has traditionally played a role in the study of the action of certain types of mappings on Hausdorff continua. It applies, for example, to collections of (i) compact nodal sets in a connected space, (ii) subcontinua of a hereditarily unicoherent continuum, and (iii) maximal cyclic elements in a peano continuum, [4]. It also applies to compact segments of the real line, where it overlaps a theorem of DEBRUNNER and HADWIGER [1] ; however, this special case can be treated by simpler methods than those used here. The following lemma is essentially lemma 1 of [2]. However, the proof is given here for the sake of completeness.

Weak cutpoint ordering on hereditarily unicoherent continua

1. Introduction. We present here two criteria for determining when an hereditarily unicoherent continuum admits the structure ot a partially ordered space which is closed, and has a unique minimal element (definitions below). Our interest in such partial orderings stems from the fact (Hunter [1]) that a one-dimensional compact connected semigroup with zero and unit admits such a partial ordering. Ward [5] has studied a class of partially ordered spaces, called generalized trees, which he characterized as hereditarily unicoherent continua which admit a closed, order-dense partial with unique minimal element, We improve his characterization by replacing order dense by the weaker monotone, and we give two other characterizations, one of which is completely topological. 2. Definitions. A partial < on a space X is a relation on X (a subset of X XX) which is reflexive, transitive, and antisymmetric. We assume throughout that X is a Hausdorff space, but not necessarily metrizable. The term arc is used to denote a continuum irreducibly connected between two points. For xCX we set L(x) = {yX:y<x}, and for ACX, L(A)=U{L(x):xCA}. We say that < is monotone if L(x) is connected for each xEX or continuous

On Unicoherency About a Simple Closed Curve

1. The subject matter of this article is a discussion of a property of point-sets which is closely related to the properties of being connected and of being a unicoherent continuum, and is in a sense a generalization of both these properties. A unicoherent continuum is defined by Kuratowski t as one which cannot be expressed as the union of two continua whose divisor is not connected, and this definition is in common usage. The analogy of the definition to that of a continuum is apparent. Ilowever, a set that is not connected may be connected between some pair of its points. We therefore propose a corresponding definition of unicoherency. A set 11 in a imetric space is unicoherent about the simple closed curve J if, for every decomposition of J into closed arcs h and k by points a and b and for every decomposition of M into relatively closed sets H and K such that h C iH, k C K., and h K = IH = a + b, ther-e is a component of H K containin.g both a and b. (If M is closed or is the space itself, the word relatively is to be omitted.) It is this extension of the idea of unicoherent continuum that is to be discussed. That this may be regarded also as a natural extension of the definition of connectivity between two points is apparent when we recall that in a one-dimensional Euclidean space an interval is a sphere and two points form its frontier or the surface of the sphere. t The property of a set being unicoherent about a simple closed curve is also useful in formulating an intrinsic definition of a two-dimensional simplex, as will be seen later. Readers of II. Whitney's recent article on this subject ? will note the marked resemblance to the property of a simple closed curve being homologous to zero in a closed set, which is the basis of his article. In fact, the present article grew out of a search for purely point-set properties equivalent to Whitney's definition.