An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each ${aₙ}_{n\in \mathbb{N}}conA⁺$ there are ${\lambda ₙ}_{n\in \mathbb{N}}\subseteq (0,\infty )$ and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...

An element $f$ of a commutative ring $A$ with identity element is called a von Neumann regular element if there is a $g$ in $A$ such that $f^{2}g=f$. A point $p$ of a (Tychonoff) space $X$ is called a $P$-point if each $f$ in the ring $C\left(X\right)$ of continuous real-valued functions is constant on a neighborhood of $p$. It is well-known that the ring $C\left(X\right)$ is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case $X$ is called a $P$-space. If all but at most one point of $X$ is a $P$-point, then $X$ is called...

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $\left|Y\right|\le 2^{{\omega }_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $\left|Y\right|\le 2^{𝔠}$ is scattered, then...

We show that the ideal of nowhere dense subsets of rationals cannot be extended to an analytic P-ideal, $F_{\sigma }$ ideal nor maximal P-ideal. We also consider a problem of extendability to a non-meager P-ideals (in particular, to maximal P-ideals).