
The mathematical community is notorious for knowing that a solution to a problem exists without actually knowing how to obtain a solution. The metrizability of GO-spaces is a case in point. Let X be a GO-space constructed on a LOTS (Y,\(\tau)\), let \(I=\{x\in X:\{x\}\in \tau \}\), \(R=\{x\in X:[x,\to)\in \tau \}\), \(L=\{x\in X:(\leftarrow,x]\in \tau \}\) and let \(D=I\cup R\cup L\). It is known that a GO-space is metrizable if and only if D is \(\sigma\)-discrete. The author constructs a metric for a countable GO-space, which, if D is dense, depends only on the enumeration of the points of X and the ordering of X. More generally, assuming that the \(\sigma\)-discrete subsets I, R, and L of a metrizable GO-space are realized as countable nested unions of discrete sets \(I=\cup_{n\in N}In\), \(R=\cup_{n\in N}Rn\), and \(L=\cup_{n\in N}Ln\), the author constructs a metric, which, if D is dense, depends only on the ordering of X and the intersection of intervals with members of \(\{Rn\}_{n\in N,}\{Ln\}_{n\in N}\) and \(\{In\}_{n\in N}\). The construction of the metrics is well motivated and well illustrated, and, although the use of both \(\emptyset\) and \(\Phi\) to denote the empty set is unfortunate, the paper on the whole is easy to follow.
metrizability of GO-spaces, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, Metric spaces, metrizability
metrizability of GO-spaces, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, Metric spaces, metrizability
