We shall prove that under CH every regular meta-Lindelöf $P$-space which is locally $D$ has the $D$-property. In addition, we shall prove that a regular submeta-Lindelöf $P$-space is $D$ if it is locally $D$ and has locally extent at most ${\omega }_1$. Moreover, these results can be extended from the class of locally $D$-spaces to the wider class of $𝔻$-scattered spaces. Also, we shall give a direct proof (without using topological games) of the result shown by Peng [On spaces which are D, linearly D and transitively D, Topology...

Let ${\left\{X_{\alpha }\right\}}_{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda$. Suppose $X$ is the quotient space of the disjoint union of $X_{\alpha }$’s by identifying $x_{\alpha }$’s as one point $\sigma$. We try to characterize ideals of $C\left(X\right)$ according to the same ideals of $C\left(X_{\alpha }\right)$’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there...

The set of isolated points (resp. $P$-points) of a Tychonoff space $X$ is denoted by $Is\left(X\right)$ (resp. $P\left(X\right))$. Recall that $X$ is said to be scattered if $Is\left(A\right)\ne ⌀$ whenever $⌀\ne A\subset X$. If instead we require only that $P\left(A\right)$ has nonempty interior whenever $⌀\ne A\subset X$, we say that $X$ is SP-scattered. Many theorems about scattered spaces hold or have analogs for SP-scattered spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also...

We show that every subgroup of an $ℝ$-factorizable abelian $P$-group is topologically isomorphic to a closed subgroup of another $ℝ$-factorizable abelian $P$-group. This implies that closed subgroups of $ℝ$-factorizable $P$-groups are not necessarily $ℝ$-factorizable. We also prove that if a Hausdorff space $Y$ of countable pseudocharacter is a continuous image of a product $X={\prod }_{i\in I}{X}_i$ of $P$-spaces and the space $X$ is pseudo-${\omega }_1$-compact, then $nw\left(Y\right)\le {\aleph }_0$. In particular, direct products of $ℝ$-factorizable $P$-groups are $ℝ$-factorizable and...

The properties of $ℝ$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $ℝ$-factorizable if and only if $G$ is $\sigma$-compact. It is proved that a subgroup $H$ of an $ℝ$-factorizable group $G$ is $ℝ$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $ℝ$-factorizable group need not be $ℝ$-factorizable, and we present a method for constructing non-$ℝ$-factorizable dense subgroups of a special class of $ℝ$-factorizable groups. Finally, we construct a closed...

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences and...