Filament sets and homogeneous continua

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.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.     

On continua with homotopically fixed boundary

The paper presents two subcontinua of Rn, one Peano-continuum, and one cellular continuum with trivial fundamental group. Both of them have the remarkable property that neither the entire spaces nor (roughly speaking) any part of them is homotopy equivalent to a lower-dimensional space. This extends work of the last three authors and of Karimov from the planar case to the higher-dimensional case, but it also contains in the cellular case the first example with all these properties in dimension two.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.     

Arc-smoothness in hyperspaces

Let X be a metric continuum and 2^x (C(X)) denote the hyperspace of closed subsets (subcontinua) of X. The concept of arc-smoothness, which is a special type of contractibility, is investigated in 2^x and C(X). Results are obtained about hyperspaces of locally connected continua, about continua for which C(X) and the cone over X are homeomorphic, about Whitney levels in C(X), and about hyperspaces of hereditarily indecomposable continua. Some examples are given and several natural questions are raised.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

Making holes in hyperspaces

Let X be a metric continuum and C(X) the hyperspace of all nonempty subcontinua of X. Let A∈C(X), A is said to make a hole in C(X), if C(X)−{A} is not unicoherent. In this paper we study the following problem.Problem: For which A∈C(X), A makes a hole in C(X).In this paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set; (3) A is a free simple closed curve; (4) A=X.     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

Cosheaves and connectedness in formal topology

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π₀ together with certain additional structure and properties that constrain π₀ to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. A cosheaf is the connected components cosheaf for a locally connected formal topology iff its complete spread is a homeomorphism, and in this case it is a terminal cosheaf.A new, geometric proof is given of the topos-theoretic result that a cosheaf is a connected components cosheaf iff it is a “strongly terminal” point of the symmetric topos, in the sense that it is terminal amongst all the generalized points of the symmetric topos. It is conjectured that a study of sites as “formal toposes” would allow such geometric proofs to be incorporated into predicative mathematics.     

Cells in hyperspaces

Let C(X) denote the hyperspace of subcontinua of a continuum X. For A∈C(X), define the hyperspace . Let k∈N, k?2. We prove that A is contained in the core of a k-od if and only if C(A,X) contains a k-cell.     

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.     

Brush spaces and the fixed point property

We introduce the notions of a brush space and a weak brush space. Each of these spaces has a compact connected core with attached connected fibers and may be either compact or non-compact. Many spaces, both in the Hausdorff non-metrizable setting and in the metric setting, have realizations as (weak) brush spaces. We show that these spaces have the fixed point property if and only if subspaces with core and finitely many fibers have the fixed point property. This result generalizes the fixed point result for generalized Alexandroff/Urysohn Squares in Hagopian and Marsh (2010) [4]. We also look at some familiar examples, with and without the fixed point property, from Bing (1969) [1], Connell (1959) [3], Knill (1967) [7] and note the brush space structures related to these examples.     

Topological characterizations of linearly uniformizable spaces

The spaces having uniformities with a totally ordered base are characterized in bigger classes of non-archimedean spaces and suborderable spaces. Consequently, several new metrization results are obtained. By examples, we show that the conditions used in our main theorem cannot be weakened essentially. Our examples may be interesting elsewhere, too.     

Homotopical smallness and closeness

The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of universal covering spaces for non-semi-locally simply connected spaces, in particular to the properties of being homotopically Hausdorff and homotopically path Hausdorff. The definitions of notions in question and their role in homotopy theory are supplemented by examples, extensional classifications, universal constructions and known applications.     

Two-dimensional convexities are join-hull commutative

It is shown that two-dimensional convex structures with certain natural properties are join-hull commutative. The main intermediate step is the computation of the so-called exchange number. We also give two examples of three-dimensional convexities which are not join-hull commutative. The second one has certain additional properties showing that the main theorem is sharp in many other respects. These properties are obtained from a study of convex hyperspaces.     

Compact Hausdorff fuzzy topological spaces are topological

Many examples of compact fuzzy topological spaces which are highly non topological are known [5, 6]. Equally many examples of Hausdorff fuzzy topological spaces which are highly non topological can be given. In this paper we show that the two properties - compact and Hausdorff - combined however necessarily imply that the fuzzy topological space is topological. This at once solves some open questions with regard to the compactification of fuzzy topological spaces [8]. It also emphasizes once more the particular role played by compact Hausdorff topological spaces not only in the category of topological spaces but even in the category of fuzzy topological spaces.     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

A lambda-dendroid with two shore points whose union is not a shore set

A subset of a given continuum is called a shore set if there is a sequence of continua in the complement of this set converging to the whole continuum with respect to the Hausdorff metric. A point is called a shore point if the one point set containing this point is a shore set. We present several examples of a lambda-dendroid which contains two disjoint shore continua whose union is not a shore set. This answers a question of Van C. Nall in negative.