Definition of “unique strongly radial convergence”

Determine a precise and coherent definition of “unique strongly radial convergence” within the terminology that uses “strongly pseudoradial” for pseudo-C-radial spaces and “unique strongly pseudoradial convergence” for the property that every continuous transfinite sequence has a unique limit. The definition should specify what uniqueness of limits for transfinite sequences (not necessarily continuous) entails in the corresponding “strongly radial” framework so that the terminology is consistent and unambiguous.

Background

The paper refactors terminology used in the literature: Brazas and Fabel employ “strongly pseudoradial” and “unique strongly pseudoradial convergence,” while this manuscript adopts “pseudo-C-radial” and “unique C‑radial convergence (UCR)” to align with the standard notion of radially closed sets and the radial property.

Within that refactored framework, there is an ambiguity if one retains the phrase “unique strongly pseudoradial convergence” but seeks an analogous notion for radial (i.e., possibly non‑continuous) transfinite sequences. The authors explicitly note that it is unclear how to define “unique strongly radial convergence,” highlighting a definitional gap in the terminology and motivating a precise formulation.