Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,{\tau }_{\delta })$ of continuous real-valued functions on the space $(X,{\tau }_{\delta })$, where ${\tau }_{\delta }$ is the smallest Tikhonov topology on X for which $\tau \subseteq {\tau }_{\delta }$ and $C(X,{\tau }_{\delta })$ is von Neumann regular. The compact and metric spaces for which $G\left(X\right)=C(X,{\tau }_{\delta })$ are characterized. Necessary, and different sufficient, conditions...

A space $X$ is functionally countable if $f\left(X\right)$ is countable for any continuous function $f:X\to ℝ$. We will call a space $X$ exponentially separable if for any countable family $ℱ$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap 𝒢\ne \varnothing$ whenever $𝒢\subset ℱ$ and $\bigcap 𝒢\ne \varnothing$. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...