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...