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...

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a $C^{*}$-point....

We present an example of a complete ${\aleph }_0$-bounded topological group $H$ which is not $ℝ$-factorizable. In addition, every $G_{\delta }$-set in the group $H$ is open, but $H$ is not Lindelöf.

Let $X$ be a Hausdorff space and let $\mathscr{H}$ be one of the hyperspaces $CL\left(X\right)$, $𝒦\left(X\right)$, $ℱ\left(X\right)$ or ${ℱ}_n\left(X\right)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathscr{H}$: extremal disconnectedness, being a $F^{\text{'}}$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $𝒦\left(X\right)$ is hereditarily disconnected. We also...

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).

It is well known that every $ℝ$-factorizable group is $\omega$-narrow, but not vice versa. One of the main problems regarding $ℝ$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega$-narrow group is a continuous homomorphic image of an $ℝ$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $ℝ$-factorizable...

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the $G_{\delta }$ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf...

If $X$ is a Tychonoff space, $C\left(X\right)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C\left(X\right)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\overline{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $gen\left(I\right)<a$ then $I$ is a non-essential ideal. We show that $X$ is an ${\aleph }_0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an ${\aleph }_1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F\left(X\right)$ denote the socle of $C\left(X\right)$. For a topological space $X$ with only...