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

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 characterization 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∈C(X):K⊂κ(p)−p} tending to X in the Vietoris topology. This contrasts with the standard characterization 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.