For a complete probability space (Ω,∑,P), the set of all complete sub-σ-algebras
of ∑, S(∑), is given a natural metric and studied. The questions of when S(∑) is compact or
connected are awswered and the important subset consisting of all continuous sub-σ-algebras is
shown to be closed. Connections with Christensen's metric on the von Neumann subalgebras of a
Type II1-factor are briefly discussed.