In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $ℐ$-quasinormal convergence. We then introduce the notion of $ℐQN\left(ℐwQN\right)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $ℐ$-quasinormally convergent to $0$ (has a subsequence which is $ℐ$-quasinormally convergent to $0$) and make certain observations on those spaces.

Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C\left(X\right)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset...

Motivated by some examples, we introduce the concept of special almost P-space and show, using the reflection principle, that for every space $X$ of this kind the inequality “$\left|X\right|\le {\psi }_c{\left(X\right)}^{t\left(X\right)}$" holds.

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...

We present several sum theorems for Ohio completeness. We prove that Ohio completeness is preserved by taking σ-locally finite closed sums and also by taking point-finite open sums. We provide counterexamples to show that Ohio completeness is preserved neither by taking locally countable closed sums nor by taking countable open sums.