On lifting properties for confluent mappings

Known results about lifting of paths for covering, light open and light confluent mappings are in some sense extended for all confluent mappings with the domain being a continuum having the arc property of Kelley. As an application we prove that each confluently tree-like continuum has the fixed point property.

     

Fans with the property of Kelley

Fans having the property of Kelley are characterized as the limits of inverse sequences of finite fans with confluent bonding mappings and as smooth fans for which the set of their end-points together with the top is closed. Also a characterization is obtained of smooth fans with the set of end-points closed as the limits of inverse sequences of finite fans with open bonding mappings.     

Some functions on hyperspaces of continua

Two natural functions from the hyperspace 2^X into the hyperspace C(C(X)) for an arbitrary metric continuum X are studied in this paper. Some characterizations of smoothness, local connectedness, connectedness im kleinen and of the property of Kelley of a continuum X are obtained in terms of continuity of these functions or their restrictions.     

强可收缩半群

称半群Ｓ强可收缩,若Ｓ的每个子半群都是它的一个缩回．本文逐次刻划了强可收缩的群、完全单半群、强可收缩的半格及正则半群的结构,在此基础上给出了任意强可收缩半群的结构定理．     

Monotone retractability and retractional skeletons

We prove that a countably compact space is monotonically retractable if and only if it has a full retractional skeleton. In particular, a compact space is monotonically retractable if and only if it is Corson. This gives an answer to a question of R. Rojas-Hernández and V.V. Tkachuk. Further, we apply this result to characterize retractional skeleton using a topology on the space of continuous functions, answering thus a question of the first author and a related question of W. Kubi?.     

Generalized Alexandroff-Urysohn squares and a characterization of the fixed point property

Given a Hausdorff continuum X, we introduce a topology on X×X that yields a Hausdorff continuum. We call the resulting space the Alexandroff-Urysohn square of X and prove that X has the fixed point property if and only if the Alexandroff-Urysohn square of X has the fixed point property.     

Open mappings increasing order

It is shown that an analog of Whyburn's theorem saying that open mappings do not increase order of a point of locally compact metric spaces is not true if the Menger-Urysohn order is replaced by order in the classical sense. On the other hand, this analog is true, even for a wider class of confluent mappings, under an additional condition that the mapping is light and the domain continuum is hereditarily unicoherent.

     

A non-treelike continuum that is not the 2-to-1 image of any continuum

Some thirteen years ago S. B. Nadler, Jr. and L. E. Ward, Jr., asked if any treelike continuum could be the 2-to-1 image of a continuum. In fact, it has been conjectured that the property of being treelike characterizes those continua that are not the 2-to-1 image of any continuum. But the characterization must be something else; this paper shows that many pseudo-solenoids are not the 2-to-1 image of any continuum.

     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

On the property of kelley in the hyperspace and Whitney continua

In this paper, we introduce the notion of property [K]¹ which implies property [K], and we show the following: Let X be a continuum and let ω be any Whitney map for C(X). Then the following are equivalent. (1) X has property [K]¹. (2) C(X) has property [K]¹. (3) The Whitney continuum ω⁻¹(t) (0⩽t<ω(X)) has property [K]¹.As a corollary, we obtain that if a continuum X has property [K]¹, then C(X) has property [K] and each Whitney continuum in C(X) has property [K]. These are partial answers to Nadler's question and Wardle's question ([10, (16.37)] and [11, p. 295]).Also, we show that if each continuum X_n (n=1,2,3,…) has property [K]¹, then the product ∏X_n has property [K]¹, hence C(∏X_n) and each Whitney continuum have property [K]¹. It is known that there exists a curve X such that X has property [K], but X×X does not have property [K] (see [11]).     

Property of Kelley for the Cartesian products and hyperspaces

A continuum having the property of Kelley is constructed such that neither , nor the hyperspace , nor small Whitney levels in have the property of Kelley. This answers several questions asked in the literature.

     

Fixed point property for 11-element sets

We introduce retractable points and show how this notion provides the key for a classification of all sets with 11 elements that have the fixed point property.     

Confluent and related mappings on arc-like continua—an application to homogeneity

It is shown that the pseudo-arc is the only nondegenerate arc-like continuum which is homogeneous with respect to the class of confluent (monotone, open) mappings. Further a concept of a pseudo-end point of an arc-like continuum is introduced and is applied to obtain another characterization of the pseudo-arc in terms of homogeneity with respect to a class of mappings.     

Filament sets, aposyndesis, and the decomposition theorem of Jones

Applications of the work introduced by the authors in a recent article, Filament sets and homogeneous continua, are given to aposyndesis and local connectedness. The aposyndetic decomposition theorem of Jones is generalized to spaces with the property of Kelley.

     

More remarks on Souslin properties and tree topologies

We investigate separation properties of ω₁-trees. We show that the property γ of Devlin and Shelah is equivalent to hereditary collectionwise normality. We show that monotone normality and divisibility are both equivalent to orderability. Finally we show that Souslin trees are examples of trees with property γ which are not retractable.     

A monotone version of the Sokolov property and monotone retractability in function spaces

We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X   is monotonically retractable if and only if _Cp(X)$C_p(X)$ is monotonically Sokolov. Besides, a space X   is monotonically Sokolov if and only if _Cp(X)$C_p(X)$ is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by R$\mathbb{R}$-quotient images and _Fσ$F_{\sigma }$-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X   and _Cp(X)$C_p(X)$ are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X   such that _Cp(X)$C_p(X)$ has the Lindelöf Σ-property but neither X   nor _Cp(X)$C_p(X)$ is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov.     

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.     

On retractable sets and the fixed point property

Call a subsetA of an ordered setP retractable tob P iff the map that mapsA tob and leaves all other points fixed is a retraction. We prove fixed point theorems for sets that contain a retractable set and also use this tool to study the fixed point property for products. The results in this paper show that three classical approaches to the fixed point property: irreducible points, cutsets and lexicographic sums can be viewed as special cases of the situation described above.Presented by I. Rival.This research was funded in part by ONR grant nr. N00014-89-J-1824.     

The Pompeiu problem and discrete groups

We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.