A continuum without non-block points

Abstract

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

The shore point existence problem is equivalent to the non-block point existence problem

“…is identical. Under the set-theoretic axiom Near Coherence of Filters (NCF) the Stone-Čech remainder H * of the half-line is a continuum with no coastal points [2]. From this we get the corollary.…”

“…For each continuum X with a non-coastal point p the author has shown [1] there is a sub- For X indecomposable the remaining propositions seem more difficult and the methods of Theorem 1 no longer apply. For example if X is indecomposable with (1) we cannot simply attach an arc to prove (2) as the quotient space is manifestly decomposable.…”

“…The result generalises to separable continua [1] but not to Hausdorff continua. Under the set-theoretic axiom Near Coherence of Filters the non-metric continuum H * lacks non-block points and hence lacks coastal points [2].…”