A METRIC FOR METRlZABLE GO-SPACES

Conditions which force the metrizability of GO-spaces are well known (see [Fa]). Since GO-spaces are T3-spaces and countable GO-spaces are first countable it follows that countable GO-spaces are metrizable. However it is not always apparent what a metric is for a given metrizable G)-space even if it is countable. For example the Sorgen fry line [S] restricted to the set of rational numbers or, if a < wI ' the LOTS [D,a] are both countable and, thus, metrizable but it is difficult to construct a metric for either of these spaces ([A]). In this note a metric is derived for GO-spaces. A LOTS ( = linearly ordered topological space) is a triple (X,A(~),~) where (X'~) is a linearly ordered set and A(~) is the usual open-interval topology generated by the order <. Recall that a subset A of X is order-convex if when ever a and b are in A, then each point lying between a and b is also in A. A GO-space ( = generalized ordered space) is a subspace of a LOTS (see [L]). There is an equivalent way to obtain a GO-space X by starting with a linearly-ordered set Y. Equip Y with a topology T that contains A(~) and has a base of open sets each of which is order-convex. In th~s case X is said to be constructed on Y and X = GOy(R,E,I,L) wher