A continuum is a compact connected metric space. For a continuum X, let C(X) denote the hyperspace of subcontinua of X. In this paper we construct two nonhomeomorphic fans (dendroids with only one ramification point) X and Y such that C(X) and C(Y) are homeomorphic. This answers a question by Sam B. Nadler, Jr.

Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).

For a metric continuum X, let C(X) (resp., ${2}^{X}$) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and ${2}^{f}:{2}^{X}\to {2}^{Y}$ be the induced functions given by $C\left(f\right)\left(A\right)=c{l}_{Y}\left(f\left(A\right)\right)$ and ${2}^{f}\left(A\right)=c{l}_{Y}\left(f\left(A\right)\right)$. In this paper, we prove that:
• If ${2}^{f}$ is almost continuous, then f is continuous.
• If C(f) is almost continuous and X is locally connected, then f is continuous.
• If X is not locally connected, then there exists an almost continuous function f: X → [0,1] such that...

We prove that the third symmetric product of a chainable continuum has the fixed point property.

Let $X$ be a continuum. Two maps $g,h:X\to X$ are said to be pseudo-homotopic provided that there exist a continuum $C$, points $s,t\in C$ and a continuous function $H:X\times C\to X$ such that for each $x\in X$, $H(x,s)=g\left(x\right)$ and $H(x,t)=h\left(x\right)$. In this paper we prove that if $P$ is the pseudo-arc, $g$ is one-to-one and $h$ is pseudo-homotopic to $g$, then $g=h$. This theorem generalizes previous results by W. Lewis and M. Sobolewski.

A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p,q\in X$ there exists a continuous surjective function $f:X\to X$ such that $f\left(p\right)=q$. Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$, $C\left(X\right)$, is not continuously homogeneous.

Let $X$ be a finite graph. Let $C\left(X\right)$ be the hyperspace of all nonempty subcontinua of $X$ and let $\mu :C\left(X\right)\to \mathbb{R}$ be a Whitney map. We prove that there exist numbers $0<{T}_{0}<{T}_{1}<{T}_{2}<\cdots <{T}_{M}=\mu \left(X\right)$ such that if $T\in ({T}_{i-1},{T}_{i})$, then the Whitney block ${\mu}^{-1}({T}_{i-1},{T}_{i})$ is homeomorphic to the product ${\mu}^{-1}\left(T\right)\times ({T}_{i-1},{T}_{i})$. We also show that there exists only a finite number of topologically different Whitney levels for $C\left(X\right)$.

A retractible non-locally connected dendroid is constructed.

Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.

An embedding from a Cartesian product of two spaces into the Cartesian product of two spaces is said to be factorwise rigid provided that it is the product of embeddings on the individual factors composed with a permutation of the coordinates. We prove that each embedding of a product of two pseudo-arcs into itself is factorwise rigid. As a consequence, if X and Y are metric continua with the property that each of their nondegenerate proper subcontinua is homeomorphic to the pseudo-arc, then X ×...

We show that there exists a C*-smooth continuum X such that for every continuum Y the induced map C(f) is not open, where f: X × Y → X is the projection. This answers a question of Charatonik (2000).

Let $X$ be a metric continuum. Let ${F}_{n}\left(X\right)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace ${F}_{n}\left(X\right)$ provided that the following implication holds: if $Y$ is a continuum and ${F}_{n}\left(X\right)$ is homeomorphic to ${F}_{n}\left(Y\right)$, then $X$ is homeomorphic to $Y$. In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace ${F}_{2}\left(X\right)$, and (2) let $X$ be an arcwise connected...

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