Linearly Ordered Topological Spaces

This work is devoted to the study of certain cardinality modifications of paracompactness and compactness in the setting of linearly ordered spaces. Some of the concepts treated here have previously been studied by Aquaro [l]1, Gulden [4], Kennison [5], Mansfield [6], Morita [7], and Poppe [9]. On the other hand, the concept of m-boundedness, introduced in §2, is new. Our main results (Theorems 1 and 3) establish the equivalence for linearly ordered spaces of a number of cardinality modifications of, in the first case, paracompactness, and, in the second, compactness. In each instance, this is accomplished by means of a characterization in terms of conditions imposed on the gaps of the space. In regard to Theorem 1, in which the concept of Q-gap introduced by Gillman and Henriksen [3 ] plays a crucial role, we call attention to the equivalence of m-paracompactness and the apparently much stronger condition m-full normality in the setting of linearly ordered spaces. It is also of interest to note that Theorem 3 shows the equivalence of m-compactness and m-boundedness, again in the setting of linearly ordered spaces. Novak [8] has shown this latter equivalence is not in general true for m countable, but the authors are not aware of an m-compact space which is not m-bounded for rrt larger than countable. 1. In this section, we note the equivalence, in the setting of linearly ordered spaces, of a varied collection of cardinality modifications of paracompactness. Unless otherwise indicated, tn will denote an infinite cardinal. Definition 1. The space A is said to be m-paracompact (m-metacompact, strongly m-paracompact) if and only if each open covering of A by no more than rrt sets admits as a refinement a locally finite (point finite, star finite) open covering. Definition 2 (Mansfield [6]). (Here, let rrt be any cardinal ^2.) Let a and (B be collections of subsets of a set A. (B> is called an m-star (almost m-star) refinement of Q, if and only if (B refines ffi and whenever ailCffi with I £011 ̂m and C\'M7*0 (MQX with \m\ ^rrt and MQSt(x, 03) for some xGA) there is an AE& with \Jm.CA (MQA). The space A is m-fully normal (almost m-fully normal) if and only if to each open covering of A there corresponds an open covering which m-star (almost m-star) refines it.