A characterization of a space with countable infinity

Abstract:

It is well known that, for a countable discrete space $N$, \[ |\beta N - N| = {2^{{2^{{\aleph _0}}}}}.\] So, any completely regular ${T_1}$ space $X$ with $\beta X - X| \leqq {\aleph _0}$ does not contain any infinite discrete subspace. In this paper, we characterize those completely regular ${T_1}$ spaces with countable infinity as follows: Such a space $X$ is characterized by the two properties. (a) $X$ is pseudocompact. (b) There exist a compact metric space $Y$ and a continuous map $f$ from $X$ onto $Y$ so that the subset ${Y_0} = \{ y:{f^{ - 1}}(y)$ of $Y$ is countable and $c{l_{\beta X}}{f^{ - 1}}(y) - {f^{ - 1}}(y)$ is one point whenever $y \in {Y_0}$. (In particular, for any $y$ in ${Y_0},\beta ({f^{ - 1}}(y)) - {f^{ - 1}}(y)$ is one point if $X$ is normal.)