The Shore Point Existence Problem is Equivalent to the Non-Block Point Existence Problem

TL;DR

This paper establishes the equivalence of three properties related to shore points and non-block points in Hausdorff continua, providing characterizations and implications for indecomposable continua.

Contribution

It proves the equivalence of properties involving shore points, non-block points, and coastal points in Hausdorff continua, and offers a new characterization of shore points.

Findings

Proves the equivalence of three properties in Hausdorff continua.

Provides a new characterization of shore points via nets of subcontinua.

Shows every point in an indecomposable continuum is a shore point.

Abstract

We prove the three propositions are equivalent: $(a)$ Every Hausdorff continuum has two or more shore points. $(b)$ Every Hausdorff continuum has two or more non-block points. $(c)$ Every Hausdorff continuum is coastal at each point. Thus it is consistent that all three properties fail. We also give the following characterisation of shore points: The point $p$ of the continuum $X$ is a shore point if and only if there is a net of subcontinua in $\{K \in C(X): K \subset \kappa(p) - p\}$ tending to $X$ in the Vietoris topology. This contrasts with the standard characterisation which only demands the net elements be contained in $X-p$. In addition we prove every point of an indecomposable continuum is a shore point.