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 ${ℱ}_n\left(X\right)$.

An estimate for the Novak number of a hyperspace with the Vietoris topology is given. As a consequence it is shown that this cardinal function can decrease passing from a space to its hyperspace.

We characterize those Tychonoff quasi-uniform spaces $(X,𝒰)$ for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family ${𝒦}_0\left(X\right)$ of nonempty compact subsets of $X$. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space $X$ is uniformly locally compact on ${𝒦}_0\left(X\right)$ if and only if $X$ is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show...

In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty $W_{\delta }$-subspace which is of the first category in itself.