Counting topologies

1 Statement of results Write |M| for the cardinal number of a set M and let P(M) denote the power set of M , P(M) = {X | X ⊂ M}. If |M| = κ then |P(M)| = 2 . In particular, c = 2א0 where as usual א0 = |N| and c = |R|. For any transfinite cardinal κ let κ+ denote the least cardinal number greater than κ . (For example, א0 = א1, א1 = א2, . . .) Naturally, κ < κ+ ≤ 2 . Neither κ+ = 2 nor κ+ < 2 is provable for any transfinite cardinal κ . Fix an infinite set X with |X | = κ and let T denote the family of all topologies τ on X . So we have τ ∈ T whenever τ ⊂ P(X) and X becomes a topological space by declaring a set U ⊂ X open if and only if U ∈ τ . Let us call two topologies τ1, τ2 isomorphic when the two spaces (X, τ1) and (X, τ2) are homeomorphic. Clearly, being isomorphic defines an equivalence relation on every family F ⊂ T . Let F∗ be the quotient set of F with respect to this equivalence relation. We are interested in computing the cardinal numbers |F | and |F∗| where F is the family of all τ ∈ T such that the topological space (X, τ ) has a certain property (P). Hence |F | is the total number of topologies τ on X where the space (X, τ )