Abstract
We study the space D(X) of ordered decompositions of a continuum X as a subspace of C(X)2 under the Vietoris topology. Our focus is on the number of components and arc components of D(X). The space D(X) is locally connected and locally compact but never compact unless it is empty. For X irreducible between two points we prove D(X) has exactly two arc components. For X locally connected we prove D(X) is arc-connected unless X is an arc. For X metric and hereditarily unicoherent we prove D(X) is arc-connected unless X is irreducible. For X metric with a cut point, we prove D(X) is arc-connected unless X is irreducible. We describe the metric continua that admit a decomposition (A,B) in a different arc component to (B,A) as unions of two proper indecomposable subcontinua. We construct an example of this where D(X) has at least three arc components. As a corollary, we get that if X is hereditarily decomposable and not irreducible then each decomposition (A,B) shares an arc component with (B,A).