Compactifications of quasi-uniform hyperspaces. (Q1868866)

In this paper the authors study compactifications of quasi-uniform hyperspaces. They prove among other things that for a Tychonoff space \(X\), the hyperspace \(2^{\beta X}\) is equivalent to the \({}^*\)-compactification of \(({\mathcal C}_0(X), {\mathcal P}{\mathcal N}_H)\) if and only if \(X\) is normal. From this they deduce that if \(X\) is normal and Hausdorff then \(\beta 2^X\) is equivalent to the \({}^*\)-compactification of \(({\mathcal C}_0(X), {\mathcal P}{\mathcal N}_H)\) if and only if \(2^X\) is pseudocompact.