Some Equivalences for Martin’s Axiom in Asymmetric Topology

We find some statements in the language of asymmetric topology and continuous partial orders which are equivalent to the statements κ < m or κ < p. We think of asymmetric topology as those parts of topology in which the specialization order, x ≤ y if and only if x ∈ c{y}, need not be symmetric. (See [5] for some of the motivations.) Martin’s Axiom has many equivalent statements, consequences, and variations in the literature which can be stated in topological terms. Most of the treatments we have seen so far from set-theoretic topologists assume that spaces are Hausdorff. In view of recent interest in asymmetric topology, in which even T1 spaces are a highly symmetric special case, we give some equivalences for Martin’s Axiom which utilize the terms of this field. Our reference for properties related to Martin’s Axiom is [2], and for properties related to continuous lattices we referred to [6]. Definition 1. A partially ordered set (P,≤) is upwards-ccc if any uncountable subset of P must have two distinct members which have a common upper bound in P. The cardinal m is the least cardinal such that there exists a non-empty upwards-ccc partially ordered set (P,≤) and a collection {Dα| α < m} of cofinal subsets of P such that no upwards-directed subset of P meets each Dα. It can be shown that ω1 ≤ m ≤ c. The Martin’s Axiom of the title is the statement m = c. Definition 2. A topological space is ccc if any uncountable collection of open sets has two distinct members which are not disjoint. A space is locally compact if every open set contains a compact neighborhood of each of its 2000 Mathematics Subject Classification. 03E50, 06B35, 06F30, 54A35, 54D45.