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 give sufficient and necessary conditions to be fulfilled by a filter $\Psi$ and an ideal $\Lambda$ in order that the $\Psi$-quotient space of the $\Lambda$-ideal product space preserves $T_k$-properties ($k=0,1,2,3,3\frac{1}{2}$) (“in the sense of the Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.

Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.

We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product $2^X$, where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set...