2020/07/17 by Daron Anderson, Anderson, Daron
Mathematics · #54F15 (Primary) 54G15 #54G20 (Secondary) #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54F15 #msc:54G15 #msc:54G20
paper · pdf · doi:10.48550/arxiv.2007.09234
arxiv created 2020/07/17 · arxiv updated 2020/07/21
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 ∈ C(X): K ⊂ κ(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.