“The kernel functor” $W\stackrel{k}{\to }LFrm$ from the category $W$ of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In $W$, monics are one-to-one, but not necessarily so in LFrm. An embedding $\varphi \in W$ for which $k\varphi$ is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The $W$-objects every embedding of which is KI are characterized;...

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...

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...

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...

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $\beta \left(f\right)\le {\omega }^{\xi ₁}·{\omega }^{\xi ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...

A family of subsets of a set is called a $\sigma$-topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A $\sigma$-topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect $\sigma$-topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a $\sigma$-topological version of Katětov-Tong...

Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire-$.5$ function between two comparable real-valued functions on the topological spaces that $F_{\sigma }$-kernel of sets are $F_{\sigma }$-sets.