305 Notre Dame Journal of Formal Logic Volume 24, Number 3, July 1983 The Axiom of Choice in Topology NORBERT BRUNNER* 1 Introduction and definitions In this paper we are concerned with soft applications of the axiom of choice {AC) in general topology. We define 16 properties which hold in ZF for each T 2 space, if and only if AC is true, and we investigate what implications between these axioms are provable without AC (in the presence of AC there is nothing to prove). Our results are sum marized in two diagrams. In Figure 1 the 61 valid implications are listed. Counterexamples in three models prove 188 of the possible implications to depend on the axiom of choice, as is shown in Figure 2. Some problems remain open. Our positive results are proved in ZF°, Zermelo Fraenkel set theory without the axioms AC and foundation. Our counterexamples are constructed in models of ZF (=^ ZF° + foundation). We shall use Levy's axiom MC of multiple choice: If F is a family of nonempty sets, there is a mapping /on F, such that φ Φ fix) C x and f{x) is finite for each x in F. Rubin's axiom PW asserts that the power set i°{x) of each well orderable set x is well orderable. In ZF°, AC implies MC which implies PW, and there are permutation models which show that in ZF° the implications cannot be reversed. But AC, MC, and PW are equivalent in ZF. Similar axioms are studied in [6]. If P and Q are topological properties, A(P) is the assertion that each T 2 space is P and A{P, Q) says that each T 2 space which is P also satisfies Q. While for the properties P defined below we do understand the position of A(P) in the hierarchy of choice principles, the same questions for A{P,Q) remain unanswered in many cases, although as we have already noticed A{P,Q) depends on AC in general. *The author wishes to express his gratitude to the referee and to U. Feigner (Tubingen) for their many useful suggestions. Received February 8, 1982; revised March 17, 1982