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...

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 X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

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 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...

It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity,...

The aim of the paper is to prove that in the unbounded Urysohn universal space $𝕌$ there is a functor of extension of $\Lambda$-isometric maps (i.e. dilations) between central subsets of $𝕌$ to $\Lambda$-isometric maps acting on the whole space. Special properties of the functor are established. It is also shown that the multiplicative group $ℝ\setminus \left\{0\right\}$ acts continuously on $𝕌$ by $\Lambda$-isometries.