An Alternative Description of Approach Spaces Via Approach Cores

The aim of this note is, following a suggestion from Lowen (2015), to give an isomorphic description of the topological construct App of approach spaces by means of families of [0, a]-valued functionals, which we will call approach cores, collectively measuring `to which degree a point can be considered to be an interior point of a subset'. We will also investigate the relation between these approach cores and other alternative descriptions of approach spaces such as a.o. via the upper hull and upper regular function frame, and from this derive a result on initiality in App of a source derived from the upper hull, which is a counterpart to a result from Lowen (2015) only available for the lower hull up till now.