Total paracompactness of real GO-spaces

A topological space is said to be totally paracompact (resp. totally metacompact) if every open base of it has a locally finite (resp. pointfinite) subcover. In this paper we characterize all totally paracompact GOspaces constructed on the real line. It turns out that in the class of GO-spaces on the real line, total paracompactness and total metacompactness are equivalent. Another consequence of our characterization is that totally metacompact GO-spaces on the real line are metrizable. Questions and partial results are given concerning total paracompactness in subspaces of real GO-spaces. A topological space is said to be totally paracompact [Fo] (totally metacompact) if every base of it has a locally finite subcover (point-finite subcover). R. Telgarsky and H. Kok [TK] showed that the Michael Line [M] was not totally paracompact. In [Le] A. Lelek asked if the Sorgenfrey Line [S] was totally paracompact. This question was answered negatively by J. M. O'Farrell [OF1] using a technique that showed neither the Sorgenfrey Line nor the Michael Line is totally metacompact. Since both the Sorgenfrey Line and the Michael Line are real GO-spaces the following questions naturally arise: 1. What GO-spaces on the real line are totally metacompact or even totally paracompact? 2. Is total metacompactness equivalent to total paracompactness in real GOspaces? In general the answer to Question 2 is no, since Heath's "V-space" [H] is totally metacompact and not (totally) paracompact. In this paper it will be shown that total metacompactness and total paracompactness are equivalent in real GO-spaces. Moreover, we shall completely characterize all totally paracompact real GO-spaces in Theorem 2.3. From this characterization it follows that totally paracompact real GO-spaces are metrizable. Recall that a linearly ordered topological space (=LOTS) is a linearly ordered set X equipped with the usual open interval topology. If < is the linear order on X then a subset C of X is order convex if whenever a and b are in C such that a < b, then {x E XI a < x < b} C C. A generalized ordered space (=GO-space) is a linearly ordered set equipped with a T1-topology for which there is a base consisting of convex sets. GO-spaces have been studied extensively (for example, see [BL1] or Received by the editors December 10, 1985 and, in revised form, August 14, 1986. The results in this paper were presented at the University of Southwest Louisiana Spring Topology Conference in April of 1986. This conference was sponsored in part by N.S.F. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F05, 54D18; Secondary 54E35.