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. ...

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.

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.

The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f\times g:X\to Y\times I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{\delta }$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and $D_j$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod }_{j=1}^k{D}_j\times I^{p+1-i}\right)$ is (i+1)-to-1 is a dense $G_{\delta }$-subset of $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ for each 0 ≤ i ≤ p.

Let $F$ be a closed subset of ${ℝ}^n$ and let $P\left(x\right)$ denote the metric projection (closest point mapping) of $x\in {ℝ}^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in ${ℝ}^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty$ and prove that the $i$th component $P_i\left(x\right)$ of $P\left(x\right)$ is differentiable a.e. if $P_i\left(x\right)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i\left(x\right)=x_i$.

We study some generalized metric properties on the hyperspace $ℱ\left(X\right)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $ℱ\left(X\right)$. Moreover, $X$ has a sequence of $wcs$-covers with property $\left(P\right)$ which is a point-star network if and only if so does $ℱ\left(X\right)$, where $\left(P\right)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable....

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 $n\in \omega$.

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)..................................................................................................................................................................

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

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

We study the relation between a space $X$ satisfying certain generalized metric properties and its $n$-fold symmetric product ${ℱ}_n\left(X\right)$ satisfying the same properties. We prove that $X$ has a $\sigma$-$\left(P\right)$-property $cn$-network if and only if so does ${ℱ}_n\left(X\right)$. Moreover, if $X$ is regular then $X$ has a $\sigma$-$\left(P\right)$-property $ck$-network if and only if so does ${ℱ}_n\left(X\right)$. By these results, we obtain that $X$ is strict $\sigma$-space (strict $\aleph$-space) if and only if so is ${ℱ}_n\left(X\right)$.

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...