Let $X$ be a Tychonoff space and denote by $\beta X$ its Stone-Čech compactification. We know, for example, that if $X$ is an $F$-space then $\beta X$ is an $F$-space and, therefore, in $\beta X$, the only convergent sequences are the eventually constant, i.e, $\beta X$ only admit trivial convergent sequences. Are there any other conditions and properties over $X$ to ensure that $\beta X$ only have trivial convergent sequences?
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Of course, a necessary condition is that $X$ have no non-trivial convergent sequences. If $X$ is realcompact, then that condition is also sufficient. For that, it is enough to show that for realcompact $X$, if $p \in \beta X$, then $p$ is not the limit of a non-trivial convergent sequence in $\beta X$. For if $p$ were the limit of a non-trivial sequence $\sigma$, then realcompactness implies that there is a zero-set $Z$ of $\beta X$ such that $Z \subseteq \beta X \setminus X$ and $Z \cap \sigma = \{ p \}$. But $\beta X \setminus Z$ is normal (because it is $\sigma$-compact) so a function on $\sigma \setminus \{p\}$ which assumes each of the values $0$ and $1$ infinitely often would extend continuously to all of $\beta X \setminus Z$, and, therefore, to all of $\beta X$. But such a function cannot extend continuously to $p$, a contradiction.
