In his paper "Continuous mappings on continua" [5], T. Maćkowiak collected results concerning mappings on metric continua. These results are theorems, counterexamples, and unsolved problems and are listed in a series of tables at the ends of chapters. It is the purpose of the present paper to provide solutions (three proofs and one example) to four of those problems.

Let $X$ be a zero-dimensional space and $C_c\left(X\right)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K\left(X\right)$ (resp., $C_c^{\psi }\left(X\right))$ the set of all functions in $C_c\left(X\right)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K\left(X\right)=O_c^{{\beta }_{0}X\setminus X}$ (resp., $C_c^{\psi }\left(X\right)=M_c^{{\beta }_{0}X\setminus {\upsilon }_{0}X}$), where ${\beta }_{0}X$ is the Banaschewski compactification of $X$ and ${\upsilon }_{0}X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c\left(X\right)$ is equal to $C_c^K\left(X\right)$, i.e., $M_c^{{\beta }_{0}X\setminus X}=C_c^K\left(X\right)$. By applying methods of functionally...

A function $f$ mapping the topological space $X$ to the space $Y$ is called a z-open function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f\left(H\right)$ is a neighborhood of ${cl}_Y\left(f\left(Z\right)\right)$ in $Y$. We say $f$ has the z-separation property if whenever $U$, $V$ are cozerosets and $Z$ is a zeroset of $X$ such that $U\subseteq Z\subseteq V$, there is a zeroset $Z^{\text{'}}$ of $Y$ such that $f\left(U\right)\subseteq Z^{\text{'}}\subseteq f\left(V\right)$. A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions...