A symmetric, idempotent, continuous binary operation on a space is called a mean. In this paper, we provide a criterion for the non-existence of mean on a certain class of continua which includes tree-like continua. This generalizes a result of Bell and Watson. We also prove that any hereditarily indecomposable circle-like continuum admits no mean.

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...

Locally planar Peano continua admitting continuous decomposition into pseudo-arcs (into acyclic curves) are characterized as those with no local separating point. This extends the well-known result of Lewis and Walsh on a continuous decomposition of the plane into pseudo-arcs.

A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction $r:X̃\genfrac{}{}{0pt}{}{onto}{⟶}X$ such that all fibers $r^{-1}\left(x\right)$ are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum $Y_X$ is called a continuous pseudo-fan of a compactum X if there are a point $c\in Y_X$ and a family ℱ of pseudo-arcs such that $\bigcup ℱ=Y_X$, any subcontinuum of $Y_X$ intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with...

Every l.s.cṁapping from a paracompact space into the non-empty, closed, convex subsets of a (not necessarily convex) $G_{\delta }$-subset of a Banach space admits a single-valued continuous selection provided every such mapping admits a convex-valued usco selection. This leads us to some new partial solutions of a problem raised by E. Michael.

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and...

We demonstrate that the set of topologically distinct inverse limit spaces of tent maps with a Cantor set for its postcritical ω-limit set has cardinality of the continuum. The set of folding points (i.e. points at which the space is not homeomorphic to the product of a zero-dimensional set and an arc) of each of these spaces is also a Cantor set.