Classes Of Continua Research Articles

Diagonals of the square of a continuum

Given a continuum X, let X2=X×X and Δ={(x,x)∈X2:x∈X}. We deal with the following properties of Δ in X2: the colocal connectedness, non-weak cut, non-block, and non-cut. We consider the following classes of continua: arcwise connected, locally connected, dendroids, indecomposable continua and metric compactifications of the ray [0,∞).

Compact spaces homeomorphic to their respective squares

We deal with topological spaces homeomorphic to their respective squares. Primarily, we investigate the existence of large families of such spaces in some subclasses of compact metrizable spaces. As our main result we show that there is a family of size continuum of pairwise non-homeomorphic compact metrizable zero-dimensional spaces homeomorphic to their respective squares. This answers a question of Włodzimierz J. Charatonik. We also discuss the situation in the classes of continua, Peano continua and absolute retracts.

On weakly continuum-chainable and almost continuum-chainable continua

We continue the study of semi-weakly continuum-chainable continua, weakly continuum-chainable continua and continuum-chainable composants. We introduce two new classes of continua, namely, semi-almost continuum-chainable and almost continuum-chainable continua and a new type of composants named union composants, which are related to continuum-chainable composants, but they are different and we give an example showing this.

Continua whose hyperspace of subcontinua is infinite dimensional and a cone

We determine several classes of continua whose hyperspaces of subcontinua are infinite dimensional and homeomorphic to cones over (usually) other continuum. In particular, we obtain many Peano continua with such a property.

Some decomposable continua and Whitney levels of their hyperspaces

We introduce the new class of continua; D⁎⁎-continua. The classes of Wilder continua and D⁎-continua are strictly contained in the class of D⁎⁎-continua. Also, the class of D-continua properly contains the class of D⁎⁎-continua. Using D⁎⁎-continua, we give the negative answer to [2, Question 2]. Furthermore, we prove that being a Wilder continuum, being a D-continuum, being a D⁎-continuum and being a D⁎⁎-continuum are Whitney properties.

Graph‐like compacta: Characterizations and Eulerian loops

AbstractA compact graph‐like space is a triple , where is a compact, metrizable space, is a closed zero‐dimensional subset, and is an index set such that . New characterizations of compact graph‐like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph‐like compacta.

Embedding suspensions into hyperspace suspensions

We present sufficient conditions to guarantee that the topological suspension, Sus(X), of a continuum X can be embedded in the hyperspace suspension, HS(X), of X in such a way that the vertexes of Sus(X) are sent to the two distinguished points of HS(X). We characterize several classes of continua having this property.

Formula omitted]-Homogeneous suspensions

A space is said to be 1 2 -homogeneous provided there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. Certain conditions on a compactum X are known under which the cone over X is 1 2 -homogeneous (Nadler Jr. and Pellicer-Covarrubias (2007) [16]). In this paper we investigate 1 2 -homogeneity in suspensions of compact metric spaces; namely, we determine necessary and sufficient conditions under which the suspension of a continuum X is 1 2 -homogeneous, among certain classes of continua. This paper gives a partial answer to a question posed by S.B. Nadler Jr. (2007) in [14, p. 342].

Non-metric Rim-metrizable continua and unique hyperspace

A class Λ of continua is said to be C-determined provided that if X, Y _ Λ and C(X) ≈ C(Y), then X ≈ Y. A continuum X has unique hyperspace provided that if Y is a continuum and C(X) ≈ C(Y), then X ≈ Y. In the realm of metric continua the following classes of continua are known to have unique hyperspace: hereditarily indecomposable continua, smooth fans (in the class of fans) and indecomposable continua whose proper and non-degenerate subcontinua are arcs. We prove that these classes have unique hyperspace in the realm of rim-metrizable non-metric continua. .

Spans of simple triods

We calculate and estimate the spans of objects in certain classes of continua; our main interest is in those objects with positive span.

Aposyndetic Continua as Bundle Spaces

Let $\mathcal {S}$ be the $P$-adic solenoid bundle, and let $\eta :X \to {S^1}$ be a map of the continuum $X$ onto ${S^1}$. The bundle space $B$ of the induced bundle ${\eta ^{ - 1}}\mathcal {S}$ is investigated. Sufficient conditions are obtained for $B$ to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.

Aposyndetic continua as bundle spaces

Let S \mathcal {S} be the P P -adic solenoid bundle, and let η : X → S 1 \eta :X \to {S^1} be a map of the continuum X X onto S 1 {S^1} . The bundle space B B of the induced bundle η − 1 S {\eta ^{ - 1}}\mathcal {S} is investigated. Sufficient conditions are obtained for B B to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.

Continuum models of porous media

The continuum theory of porous materials respecting the internal discontinuity of these materials, is formulated in this paper. It is shown that porous material can be studied in two classes of continua (continua associated with volume and mass average quantities), five different kinds existing in each class. The relation of these classes and kinds to various parts and configurations of the porous medium is defined. Transformation equations between the classes and kinds are determined. The value of the principle of superposition of “components” is discussed and the principle of superposition of states is formulated.

Quasi-monotone images of certain classes of continua

Let ƒ be a continuous map from a compact metric continuum X onto a continuum Y. Then ƒ is quasi-monotone if, for each subcontinuum K of Y with nonvoid interior, ƒ -1( K) has a finite number of components and each is mapped onto K by ƒ. Examples of quasi-monotone maps are local homeomorphisms and other finite to one confluent maps. In the following all maps are assumed to be quasi-monotone from X onto Y. A theorem of L. Mohier and J.B. Fugate [1] says that if X is irreducible between two of its points then Y is also irreducible between two of its points. This result is generalized to the following theorem. If X is irreducible about a finite point set A then either Y is irreducible about ƒ( A) or there is a point y in Y such that Y is irreducible about { y}⋃ƒ( A⧹{α}) for each a in A. Another result is that if X is a continuum that is separated by no subcontinuum, i.e., a θ 1-continuum, then Y is a θ 1-continuum or is irreducible between two of its points.