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).

We prove several stability properties for the class of compact Hausdorff spaces $T$ such that $C\left(T\right)$ with the weak or the pointwise topology is in the class of Stegall. In particular, this class is closed under arbitrary products.

We give a partial classification of spaces $C_p\left([1,\alpha ]\right)$ of continuous real valued functions on ordinals with the topology of pointwise convergence with respect to homeomorphisms and uniform homeomorphisms.

We apply the general theory of $\tau$-Corson Compact spaces to remove an unnecessary hypothesis of zero-dimensionality from a theorem on polyadic spaces of tightness $\tau$. In particular, we prove that polyadic spaces of countable tightness are Uniform Eberlein compact spaces.

We show that a completely regular space Y is a p-space (a Čech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.

A space $X$ is functionally countable (FC) if for every continuous $f:X\to ℝ$, $\left|f\right(X\left)\right|\le \omega$. The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindelöf P-spaces, $\sigma$-products in $2^{\kappa }$, and some L-spaces. We consider the following three versions of functional separability: $X$ is 1-FS if it has a dense FC subspace; $X$ is 2-FS if there is a dense subspace $Y\subset X$ such that for every continuous $f:X\to ℝ$, $\left|f\right(Y\left)\right|\le \omega$; $X$ is 3-FS if for every continuous $f:X\to ℝ$, there is a dense subspace $Y\subset X$ such that $\left|f\right(Y\left)\right|\le \omega$. We give examples distinguishing...

The concept of the distinguished sets is applied to the investigation of the functionally countable spaces. It is proved that every Baire function on a functionally countable space has a countable image. This is a positive answer to a question of R. Levy and W. D. Rice.

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...