A note on the construction of non-spatial frames

Abstract

We introduce a new type of ideal, which we term a ζ-ideal (zeta-ideal), on a topological space. We present examples of ζ-ideals and study their properties. ζ-Ideals are shown to provide a new paradigm for constructing non-spatial frames. Beyond that, we map out an intrinsic design of a process that removes the given points from an arbitrary topological space to obtain a frame without those points. We also define the concept of a kernel of a ζ-ideal and characterise spatiality of newly constructed frames using ζ-ideals and their kernels.