CONTENTSIntroduction.......................................................................................................51. Preliminaries.................................................................................................6  A. Mappings....................................................................................................6  B. Arc-like continua.........................................................................................8  C. Pseudosuspensions...................................................................................8  D....

Let $X$ be a continuum and $n$ a positive integer. Let $C_n\left(X\right)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$, define the hyperspace ${C_n}_K\left(X\right)=\{A\in C_n\left(X\right):K\subset A\}$. In this paper, we consider the hyperspace $C_K^n\left(X\right)=C_n\left(X\right)/{C_n}_K\left(X\right)$, which can be a tool to study the space $C_n\left(X\right)$. We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility. ...

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.

We show that if n is a positive integer and $2^{\aleph ₀}\le \aleph ₙ$, then for every positive integer m and for every real constant c > 0 there are functions $f₁,...,f_{n+m}:ℝⁿ\to ℝ$ such that $(f₁,...,f_{n+m})\left(ℝⁿ\right)={ℝ}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i₁},...,f_{iₙ})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {ℝ}^{n-1}$.

A connected topological space $Z$ is unicoherent provided that if $Z=A\cup B$ where $A$ and $B$ are closed connected subsets of $Z$, then $A\cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z\in Z$ makes a hole in $Z$ if $Z-\left\{z\right\}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^X$ can be extended to a Whitney map for $2^X$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

We discuss the discrete $p$-Laplacian eigenvalue problem, $$\left\{\begin{array}{c}\Delta \left({\phi }_p\left(\Delta u(k-1)\right)\right)+\lambda a\left(k\right)g\left(u\left(k\right)\right)=0,k\in \{1,2,...,T\},\hfill \\ u\left(0\right)=u(T+1)=0,\hfill \end{array}\right.$$ where $T>1$ is a given positive integer and ${\phi }_p\left(x\right):={\left|x\right|}^{p-2}x$, $p>1$. First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(\lambda ,u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $\lambda >0$.

CONTENTSIntroduction................................................................................................................................................................................3I. General properties of k-to-one functions on locally compact spaces1. Multi-valued functions Ф and ψ......................................................................................................................................... 62. The proof of (I.11)..................................................................................................................................................................

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

A new ⋄-like principle ${\diamond }_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond }_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond }_{}$ holds. As ${\diamond }_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond }_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁...