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...

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.

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.