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 II 1 -factor are briefly discussed.