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We study the shore and non-block points of non-metric continua. We reduce the problem of showing a continuum to have non-block points to that of showing an indecomposable continuum to have non-block points. As a corollary we prove that separable continua have at least two nonblock points-and moreover are irreducible about their set of non-block points.

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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 monotone 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. Bellamy who constructed the first example in ZFC [3]. There are very few examples known. The Daron Anderson 1 No Non-Block Points a continuum has a coastal point if and only if it has a non-block point, f and only if it contains a

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