Shore and non-block points in Hausdorff continua

Abstract

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.

A continuum without non-block points

“…Condition (1) shows that when D is a Q-point and f : ω → ω finite-to-one then f (D) is either principal or is a permutation of D. The next lemma proves our assertion that Q-points are the ≤-minimal ultrafilters.…”

Ultracoproduct continua and their regular subcontinua

“…Theorem 5.1 says that a metrizable continuum is not only coastal at each of its points, but coastal at each of its proper subcontinua (in the obvious broader sense). In the interests of extending this result to all continua, the following "reduction" theorem is an immediate consequence of the techniques developed in, 1 and is a minor improvement on [ We prove the following as we did Proposition 4.1.…”

Separable Indecomposable Continuum with Exactly One Composant

“…To close the section we remark that metrisability cannot be droped from the hypothesis of Lemma 5.2. Example 5.8 of [1] is the closed unit ball in the Hilbert space ℓ 2 (ω 1 ) of squaresummable functions ω 1 → R under the weak topology. In fact ℓ 2 (ω 1 ) is even a continuum and the elements of the ω 1 -chain are nowhere dense subcontinua.…”