On the weight and far points of compactifications (Q1107871)

The following results are proved: (i) Let \(\tau\) be an uncountably cofinal cardinal and X be a metric space of weight \(\tau\). Then the weight of every metric extension of X is equal to \(2^{\tau}\). Here a compactification bX of a metric space X is called a metric extension of X if there is a metric d compatible with the topology of X such that \(d(F,G)>0\) if \(cl_{bX}(F)\cap cl_{bX}(G)=\emptyset\) for each two subsets F and G of X; (ii) Let bX be a metric extension of X and \(X^*=bX-X\). Then every \(x\in X\) belongs to the closure of some discrete subset of X. If X is a countable discrete sum of compacta, \(X^*\) does not contain remote points: (iii) Let X be a first countable non- pseudocompact space with no isolated points. Then there is a remote point \(x\in \beta X-X\) which belongs to the closure of some discrete subset of X.