A continuum without non-block points

For any composant E⊂H⁎ and corresponding near-coherence-class E⊂ω⁎ we prove the following are equivalent: (1) E properly contains a dense semicontinuum. (2) Each countable subset of E is contained in a dense proper semicontinuum of E. (3) Each countable subset of E is disjoint from some dense proper semicontinuum of E. (4) E has a minimal element in the finite-to-one weakly-increasing order of ultrafilters. (5) E has a Q-point. A consequence is that NCF is equivalent to H⁎ containing no proper dense semicontinuum and no non-block points. This gives an axiom-contingent answer to a question of the author. Thus every known continuum has either a proper dense semicontinuum at every point or at no points. We examine the structure of indecomposable continua for which this fails, and deduce they contain a maximum semicontinuum with dense interior.