What does connected components mean in the context of non-graphs? Graphs have vertices and the vertices are connected by edges. Hence, you can build a spanning tree (for example) by systematically joining connected components (where connected components refer to connected subgraphs).
Previous discussion on this topic reveals that each connected component of a linear decision tree on some function F represents a particular region bounded by a set of half-planes and hyperplanes. For linear decision trees applied to the element uniqueness problem, we have 2 possible type of connected components (correct me if I'm wrong) - YES and NO components.
Apparently there is only 1 NO component, and at least n! YES components when for linear decision trees applied to the element uniqueness problem on a set of "n" distinct items. I can understand the number of YES components (simply permute all items), but I'm not sure why there's only 1 NO component.