The metrization of upper limit topologies (Q752532)

The author considers a class of GO-spaces obtained as follows: If \((X,<)\) is a linear ordered set then \(X_ u\) denotes X with the topology which has the collection of all intervals ]a,b[, a,b\(\in X\), \(a<b\), as a basis. The main result is the following: Theorem 4. The space \(X_ u\) is metrizable if and only if it is the union of countably many discrete closed subspaces. Reviewer's remarks: (1) If \((X,<)\) has the least element \(a_ 0\) then \(\{a_ 0\}\) should be assumed to belong to the basis of \(X_ u\). (2) Theorem 4 is an immediate corollary to Theorem 3.1 and 3.2 proved by \textit{M. J. Faber} in [Metrizability in generalized ordered spaces, Math. Centre Tracts, 53 (1974; Zbl 0282.54017)].