Clan acts and codimension

Let (T, X) be a continuum act, let cd X=n and suppose A is a T-ideal (i.e., a T-invariant subspace of X), such that Hⁿ(A)≠0. We prove that A is a minimal T-ideal iff A=Gx for some x∈X and maximal group G in the minimal ideal of T. Moreover, if these conditions are satisfied, then A is the only minimal T-ideal and also is the unique floor for every nonzero element of Hⁿ(X). We need and also prove here an improved version of the Tube Theorem 3], and this corollary: if (G, X) is an intransitive transformation group with G compact, X locally compact and finite dimensional, and X/G connected, then dimension Gx<dimension X for all x∈X. NSF GP 9659. NSF GP 28655.