On w-unicoherence, top-irreducibility and Whitney levels near the top

Given a metric continuum X, we consider C(X) as the collection of all subcontinua of X. S. López introduced the concepts of pseudo-linearity and pseudo-circularity in order to characterize continua having a positive Whitney level that is an arc or a simple closed curve. In this paper, we introduce the concepts of w-unicoherence and top-irreducibility. We study the relations between these and the well-known and more naturally-related properties defined in continuum theory, and with the concepts of pseudo-linearity and pseudo-circularity. Moreover, by using these new concepts, we obtain a new characterization of continua which have a positive Whitney level that is an arc or a simple closed curve. Also, we provide necessary conditions for a continuum to have a positive Whitney level which is a simple n-od.