Unicoherence at subcontinua (Q1074171)

The author introduces a localized notion of unicoherence: a metric continuum X is unicoherent at a subcontinuum Y provided if A and B are proper subcontinua of X and \(X=A\cup B\) then \(A\cap B\cap Y\) is connected. General results are obtained and used to characterize two other forms of unicoherence. In particular, it is shown that strong unicoherence [defined by Bennett] is equivalent to unicoherence at every subcontinuum and weak hereditary unicoherence [defined by Maćkowiak] is equivalent to unicoherence at every subcontinuum with nonempty interior. Strong unicoherence and weak hereditary unicoherence are proved equivalent for continua in which each indecomposable subcontinuum has nonempty interior. Finally, several characterizations of dendrites are obtained.