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We show, in particular, that every remote point of $X$ is a nonnormality point of $\beta X$ if $X$ is a locally compact Lindelöf separable space without isolated points and $\pi w\left(X\right)\le {\omega }_1$.

$\beta X-\left\{p\right\}$ is non-normal for any metrizable crowded space $X$ and an arbitrary point $p\in X^{*}$.

Totally nonremote points in $\beta \mathbb{Q}$ are constructed. The number of these points is $2^{𝔠}$.

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.

J. Terasawa in "$\beta X-\left\{p\right\}$ are non-normal for non-discrete spaces $X$" (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech–Stone remainder $X^{*}$ is a non-normality point of $\beta X$. We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

Remote points constructed so far are actually strong remote. But we construct remote points of another type.

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...

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.

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.