We recall a characterization of hereditary indecomposability originally obtained by Krasinkiewicz and Minc, and show how it may be used to give unified constructions of various hereditarily indecomposable continua. In particular we answer a question asked by Mackowiak and Tymchatyn by showing that any continuum of arbitrary weight is a weakly confluent image of a hereditarily indecomposable continuum of the same weight.
We present two methods of constructing these preimages: (a) by model-theoretic means, using the compactness and completeness theorems from first-order logic to derive these results for continua of uncountable weight from their metric counterparts; and (b) by constructing essential mappings from hereditarily indecomposable continua onto Tychonoff cubes.
We finish by reviving an argument due to Kelley about hyperspaces of hereditarily indecomposable continua and show how it leads to a point-set argument that reduces Brouwer's Fixed-point theorem to its three-dimensional version.
A continuum $K$ is a common model for the family ${\mathcal K}$ of continua if every member of ${\mathcal K}$ is a continuous image of $K$. We show that none of the following classes of spaces has a common model: 1) the class of strongly chaotic hereditarily indecomposable $n$-dimensional Cantor manifolds, for any given natural number $n$, 2) the class of strongly chaotic hereditarily indecomposable hereditarily strongly infinite-dimensional Cantor manifolds, 3) the class of strongly chaotic hereditarily indecomposable continua with transfinite dimension (small or large) equal to $α$, for any …
Continuously homogeneous hereditarily indecomposable continua are tree-like
Jagiellonian University in Kraków（克拉科夫雅盖隆大学）；National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling（IT4创新国家超级计算中心，俄斯特拉发大学分部，模糊建模研究与应用研究所）；Gannon University（甘农大学）；Auburn University（奥本大学）
We give a complete description of the structure of the connected components of the general linear group of a real hereditarily indecomposable Banach space, depending on the existence of complex structures on the space itself and on its hyperplanes. A side result is the fact that complex structures cannot exist simultaneously on such a space and on its hyperplanes.
A continuum is hereditarily equivalent if it is homeomorphic to each of its non-degenerate sub-continua. We show in this paper that the arc and the pseudo-arc are the only non-degenerate hereditarily equivalent plane continua.
Given a continuum $X$, let $\mathcal{NB} (\mathcal{F}_1(X))$ be the hyperspace of nonblockers of $\mathcal{F}_1(X)$. In this paper, we show that if $X$ is hereditarily decomposable with the property of Kelley such that $\mathcal{NB} (\mathcal{F}_1(X))$ is a continuum, then $X$ is a simple closed curve. Thus, we characterize the simple closed curve as the unique hereditarily decomposable continuum with the property of Kelley $X$ such that its hyperspace $\mathcal{NB} (\mathcal{F}_1(X))$ is a continuum.
E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if $Y$ is a hereditarily unicoherent Suslinian continuum, then there exists $\varepsilon>0$ such that every two $\…
In this paper, we consider spaces whose Higson coronae are indecomposable continua. We show that for a non-compact proper metric space $X$ which is coarsely geodesic and has coarse bounded geometry, the Higson corona of $X$ is an indecomposable continuum if and only if $X$ is coarsely equivalent to the space of natural numbers. Then we give characterizations of finitely generated groups that have one or two ends by decomposability/indecomposability of the components of their Higson coronae. we characterize it as a group whose Higson corona is a topological sum of two indecomposable continua.
Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecomposable Banach spaces always embed into $\ell_\infty$ and so their density and cardinality is bounded by the continuum and because dual Banach spaces of densities bigger than continuum are decomposabl…
Indecomposable continua with one composant are $\textit{large}$ in the sense of being non-metrisable. We adapt the method of Smith $[18]$ to construct an example which is $\textit{small}$ in the sense of being separable.
The family of Wilder continua in cubes of dimension > 2 and its two subfamilies-of continuum-wise Wilder continua and of hereditarily arcwise connected continua-are recognized as coanalytic absorbers in the hyperspace of subcontinua of the cubes. In particular, each of them is homeomorphic to the set of all nonempty countable closed subsets of the unit interval.
We construct an indecomposable continuum with exactly one strong non-cut point. The method is an adaptation of Bellamy $[1]$. We start with an $ω_1$-chain of indecomposable metric continua and retractions. The inverse limit is an indecomposable continuum with exactly two composants. Our example is formed by identifying a point in each composant.
We show that every non-degenerate homogeneous plane continuum is homeomorphic to either the unit circle, the pseudo-arc, or the circle of pseudo-arcs. It follows that any planar homogenous compactum has the form $X \times Z$, where $X$ is a either a point or one of these three homogeneous plane continua, and $Z$ is a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc: a non-degenerate continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable and has span zero.
We study the shore and non-block points of non-metric continua. We reduce the problem of showing a continuum to have non-block points to that of showing an indecomposable continuum to have non-block points. As a corollary we prove that separable continua have at least two non-block points -- and moreover are irreducible about their set of non-block points.
Knaster continua and solenoids are well-known examples of indecomposable continua whose composants (maximal arcwise-connected subsets) are one-to-one images of lines. We show that essentially all non-trivial one-to-one composant images of (half-)lines are indecomposable. And if $f$ is a one-to-one mapping of $[0,\infty)$ or $(-\infty,\infty)$, then there is an indecomposable continuum of which $X:=$ran$(f)$ is a composant if and only if $f$ maps all final or initial segments densely and every non-closed sequence of arcs in $X$ has a convergent subsequence in the hyperspace $K(X)\cup \{X\}$. We…
We prove that for a chainable continuum $X$ and every non-zigzag $x\in X$ there exists a planar embedding $ϕ:X\to ϕ(X)\subset\mathbb R^2$ such that $ϕ(x)$ is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings $ϕ,ψ:X \to \mathbb R^2$ are called strongly equivalent if $ϕ\circ ψ^{-1}: ψ(X) \to ϕ(X)$ can be extended to a homeomorphism of $\mathbb R^2$. We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.
A subset $M$ of a continuum $X$ is called a \textit{meager composant} if $M$ is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of $X$. Motivated by questions of Bellamy, Mouron and Ordoñez, we show that no tree-like continuum has a proper open meager composant, and that every tree-like continuum has either $1$ or $2^{\aleph_0}$ meager composants. We also prove a decomposition theorem: If $X$ is tree-like and every indecomposable subcontinuum of $X$ is nowhere dense, then the partition of $X$ into meager composants is upper semi-c…
Suppose $Y$ is a continuum, $x\in Y$, and $X$ is the union of all nowhere dense subcontinua of $Y$ containing $x$. Suppose further that there exists $y\in Y$ such that every connected subset of $X$ limiting to $y$ is dense in $X$. And, suppose $X$ is dense in $Y$. We prove $X$ is homeomorphic to a composant of an indecomposable continuum, even though $Y$ may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordoñez in 2016. If $Y$ is chainable or, more generally, an inverse limit of identical topological graphs, then we show $Y$ is indecomposable …
Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $φ:X\to L'$ onto a locally connected continuum is a composition $φ=φ'\circ m$ where $φ':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents…
