A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.

The set of points of upper semicontinuity of multi-valued mappings with a closed graph is studied. A topology on the space of multi-valued mappings with a closed graph is introduced.

For a topological space $X$, let $S\left(X\right)$ denote the set of all closed subsets in $X$, and let $C\left(X\right)$ denote the set of all continuous maps $f:X\to X$. A family $𝒜\subseteq S\left(X\right)$ is called reflexive if there exists $𝒞\subseteq C\left(X\right)$ such that $𝒜=\{A\in S(X):f(A)\subseteq A$ for every $f\in 𝒞\}$. Every reflexive family of closed sets in space $X$ forms a sub complete lattice of the lattice of all closed sets in $X$. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed...

Answering a question of Telgársky in the negative, it is shown that there is a space which is β-favorable in the strong Choquet game, but all of its nonempty $W_{\delta }$-subspaces are of the second category in themselves.

We calculate the density of the hyperspace of a metric space, endowed with the Hausdorff or the locally finite topology. To this end, we introduce suitable generalizations of the notions of totally bounded and compact metric space.

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.

In this paper, we generalize the classical Hausdorff metric with t-norms and obtain its basic properties. Furthermore, for a given stationary fuzzy metric space with a t-norm without zero divisors, we propose a method for constructing a generalized Hausdorff fuzzy metric on the set of the nonempty bounded closed subsets. Finally we discuss several important properties as completeness, completion and precompactness.

The Lifshits theorem states that any k-uniformly Lipschitz map with a bounded orbit on a complete metric space X has a fixed point provided k < ϰ(X) where ϰ(X) is the so-called Lifshits constant of X. For many spaces we have ϰ(X) > 1. It is interesting whether we can use the Lifshits theorem in the theory of iterated function systems. Therefore we investigate the value of the Lifshits constant for several classes of hyperspaces.

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 ${ℱ}_n\left(X\right)$.