We show that ${\omega }^{*}\setminus \left\{p\right\}$ is not normal, if $p$ is a limit point of some countable subset of ${\omega }^{*}$, consisting of points of character ${\omega }_1$. Moreover, such a point $p$ is a Kunen point and a super Kunen point.

A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.

By studying dimensional types of metric scattered spaces, we consider the wider class of metric σ-discrete spaces. Applying techniques relevant to this wider class, we present new proofs of some embeddable properties of countable metric spaces in such a way that they can be generalized onto uncountable metric scattered spaces. Related topics are also explored, which gives a few new results.

Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

The notion of quasi-p-boundedness for p ∈ ${\omega }^{*}$ is introduced and investigated. We characterize quasi-p-pseudocompact subsets of β(ω) containing ω, and we show that the concepts of RK-compatible ultrafilter and P-point in ${\omega }^{*}$ can be defined in terms of quasi-p-pseudocompactness. For p ∈ ${\omega }^{*}$, we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × $P_{RK}\left(p\right)$ is bounded in X × $P_{RK}\left(p\right)$, if and only if $c{l}_{\beta \left(X\times P_{RK}\left(p\right)\right)}\left(B\times P_{RK}\left(p\right)\right)=c{l}_{\beta X}B\times \beta \left(\omega \right)$, where $P_{RK}\left(p\right)$ is the set of Rudin-Keisler predecessors of p.

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...

We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of ${\omega }^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of ${\omega }^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space” (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs...