On barely $\alpha$-compact spaces and remote points in $\beta _ \alpha X_ X$

Abstract:

Let $X$ be a Tychonoff space. It was proved by Terada that if the cellularity of $X$ is not Ulam-measurable, then no point of $\upsilon X\backslash X$ is a remote point of $X$. In this paper we generalize this result by proving that if $X$ is barely $\alpha$-compact, then no point in ${\beta _\alpha }X\backslash X$ is an $\alpha$-exotic point of $X$. This implies, in particular, that if no cellular family in $X$ has a $\alpha$-measurable cardinality, then no point of ${\beta _\alpha }X\backslash X$ is a remote point of $X$; the Terada theorem then follows as a corollary when $\alpha = {\omega _1}$.