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

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.

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.

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.

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 $X$ 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.

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

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 $\pi w\left(X\right)\le {\omega }_1$.

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