Complements in the lattice of topologies on a set of singular cardinality

Two topologies on a set are complementary if their meet is the indiscrete topology and their join is the discrete topology. Watson (1994) determined the possible numbers of complements of a topology on a set of size ℵn. He asked what happens on a set of singular cardinality, and in particular whether a topology on ℵω can have more than 2^ℵω and more than sup_n 2^(2^ℵn) complements but fewer than 2^(2^ℵω) (his Problem 6.2). We determine the number of complements of every topology on a set of any infinite cardinality κ. Apart from the discrete topology, the indiscrete topology and the free unions of an indiscrete and a discrete space, a topology τ on κ has exactly 2^(κ·2^(<λ(τ))) complements. Here λ(τ) is the least cardinal larger than the sizes of all minimal neighbourhoods in complements of τ. The key step holds for all κ, singular ones included: if some complement has a point all of whose neighbourhoods have at least μ points, then the topology has at least 2^(2^μ) complements. We also give a self-contained proof of Watson’s structure theorem for T₀ quotients. Watson’s spaces CO(<λ, κ) have 2^(κ·2^(<λ)) complements. So his Lemma 5.11 fails at limit cardinals in some models of ZFC, and the list of values proposed in his Problem 6.1 needs correcting. Problem 6.2 has a positive answer exactly when max(2^ℵω, sup_n 2^(2^ℵn)) < 2^(2^(<ℵω)) < 2^(2^ℵω). This holds in a model obtained by Easton forcing and fails whenever ℵω is a strong limit, so the problem is independent of ZFC. On a set of singular strong limit cardinality κ, the number of complements is always 1, κ, 2^κ or 2^(2^κ).