arXiv subjects
Publications and source records attributed to Alejandro Illanes.
Given a continuum $X$, let $C(X)$ be the hyperspace of all subcontinua of $X$. We consider the hyperspace $NC^{*}(X)=\{A\in C(X):X\setminus A$ is connected$\}$. In this paper we prove that the only locally connected continua $X$ for which $NC^{*}(X)$ is compact are the arcs and the simple closed curves. We also characterize the finite graphs $G$ for which $NC^{*}(G)$ is connected.
A fan is an arcwise-connected continuum, which is hereditarily unicoherent and has exactly one ramification point. Many of the known examples of fans were constructed as 1-dimensional continua that are unions of arcs which intersect in exactly one point. Borsuk proved in 1954 that each fan is a 1-dimensional continuum which is the union of arcs intersecting in exactly one point. But it is not yet known if this property is equivalent to being a fan. In this paper, we show that under two additional assumptions, every such union of arcs is a fan.
A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^{2}$ denote the Cartesian square of $X$ and $Δ$ the diagonal of $X^{2}$. In \cite{ka} it was asked if for a continuum $X$, distinct from the arc, $X^{2}\setminus Δ$ is continuumwise connected if and only if $X$ is decomposable. In this paper we show that no implication in this question holds. For the proof of the non-necessity, we use the dynamic properties of a suitable homeomorphism of the Cantor set onto itself to construct an appropriate indecomposable continuum $X$.
Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense subset of $X$. The collection of all nonempty proper closed subspaces $B$ of $X$ such that $B$ does not block any element of $X\setminus B$ is denoted by $NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic to $NB(F_{1}(X))$. This answers a series of questions by Camargo, Capulín, Castaneda-Alvarado and Maya.
Given a collection of pairwise co-prime integers $% m_{1},\ldots ,m_{r}$, greater than $1$, we consider the product $Σ=Σ_{m_{1}}\times \cdots \times Σ_{m_{r}}$, where each $Σ_{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. Łysko, in this paper we prove that if $M$ is a subcontinuum of $Σ$ such that the projections of $M$ on each $Σ_{m_{i}}$ are onto, then for each open subset $U$ in $Σ$ with $M\subset U$, there exists an open connected subset $V$ of $Σ$ such that $M\subset V\subset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $Σ_{m_{i}}\times Σ_{m_{i}}$, whose diagonals are never ample [1].
A space X has the local fixed point property LFPP, (local periodic point property LPPP) if it has an open basis $\mathcal{B}$ such that, for each $B\in \mathcal{B}$, the closure $\overline{B}$ has the fixed (periodic) point property. Weaker versions wLFPP, wLPPP are also considered and examples of metric continua that distinguish all these properties are constructed. We show that for planar or one-dimensional locally connected metric continua the properties are equivalent.
An example is given of a compact absolute retract that is not a Hilbert cube manifold but whose second symmetric porduct is the Hilbert cube. A factor theorem is given for nth symmetric product of the cartesian product of any absolute neighborhood retract with the Hilbert cube. A short proof is included of the known fact that symmetric products preserve the property of being a compact Hilbert cube manifold (the theorem is proved here for all Hilbert cube manifolds).