Lemma 4.2

  If is connected, and the multiset is in A for some , then the subset of all coordinates of points in A is connected.

Proof

Suppose not. Then there are open sets U, such that and are disjoint nonempty sets with union B. Without loss of generality, . Let

Then , are open sets in which are stable under , so they project to open sets , in . Also since a point in A must have all coordinates in U, or else at least one coordinate in . Furthermore , and is nonempty also, since at least one point of A has a coordinate in V, since . Finally , since it is not possible for a point of A to have all coordinates in U, yet have some coordinate in V. This contradicts the connectedness of A.