In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano $X$, $Y$ spazi topologici e $C\left(X,Y\right)$ l'insieme delle funzioni continue da $X$ verso $Y$. Fissato un «network» $\alpha$ nel dominio $X$ ed una topologia $\tau$ nell'iperspazio $CL\left(Y\right)$ del codominio $Y$ si genera una topologia ${\tau }_{\alpha }$ in $C\left(X,Y\right)$ richiedendo che una rete $\left\{f_{\lambda }\right\}$ di $C\left(X,Y\right)$ converge in ${\tau }_{\alpha }$ ad $f\in C\left(X,Y\right)$ se e solo se la rete $\left\{\overline{f_{\lambda }\left(A\right)}\right\}$ converge in $\tau$ ad $\overline{f\left(A\right)}$...

Necessary conditions and sufficient conditions are given for $C_p\left(X\right)$ to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with $C_p\left(K\right)$ an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).

For non-empty topological spaces X and Y and arbitrary families $𝒜$ ⊆ $𝒫\left(X\right)$ and $ℬ\subseteq 𝒫\left(Y\right)$ we put ${𝒞}_{𝒜,ℬ}$=f ∈ $Y^X$ : (∀ A ∈ $𝒜$)(f[A] ∈ $ℬ)$. We examine which classes of functions $ℱ$ ⊆ $Y^X$ can be represented as ${𝒞}_{𝒜,ℬ}$. We are mainly interested in the case when $ℱ=𝒞(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $ℱ=𝒞$(X,ℝ) is not equal to ${𝒞}_{𝒜,ℬ}$ for any $𝒜$ ⊆ $𝒫\left(X\right)$ and $ℬ$ ⊆ $𝒫$(ℝ). Thus, $𝒞$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

Given a metric continuum $X$ and a positive integer $n$, $F_n\left(X\right)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_n\left(X\right)$, $F_n(K,X)$ denotes the set of elements of $F_n\left(X\right)$ containing $K$ and $F_n^K\left(X\right)$ denotes the quotient space obtained from $F_n\left(X\right)$ by shrinking $F_n(K,X)$ to one point set. Given a map $f:X\to Y$ between continua, $f_n:F_n\left(X\right)\to F_n\left(Y\right)$ denotes the induced map defined by $f_n\left(A\right)=f\left(A\right)$. Let $K\in F_n\left(X\right)$, we shall consider the induced map in the natural way $f_{n,K}:F_n^K\left(X\right)\to F_n^{f\left(K\right)}\left(Y\right)$. In this paper we consider the maps $f$, $f_n$, $f_{n,K}$ for some $K\in F_n\left(X\right)$ and $f_{n,K}$ for...

We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense $G_{\delta }$-subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.