  This is the last in a series of three notes on an investigation into core
regular double Stone algebras, CRDSA, which are meant to be read in order. This
note ends our initial investigation of duality for CRDSA through bi-topological
spaces. More succinctly, duality through refinement of a pre-established
duality of pairwise Stone spaces and bounded distributive lattices. In this
note we show that the pairwise Stone derived from a CRDSA L has a base that is
a CRDSA isomorphic to L. For the purposes of this note only we will call any
such pairwise zero-dimensional space a core regular double pairwise
zero-dimensional space and similarly for the corresponding pairwise Stone
spaces. Then we establish necessary and sufficient conditions for a
bi-continuous map to have an inverse that is a CRDSA homomorphism. These
results are topologically indactive of just how "nearly Boolean" CRDSA are,
these inverses must respect the appropriate conditions on the boundary of
non-clopen elements of the bases of the bi-topological space. From that result
we validate our earlier claim that the category of core regular double Stone
algebras is dually equivalent to the category of core regular double pairwise
Stone spaces. We note that the conditions for this duality should be able to be
easily relaxed to yield a duality for a less rigid subclass of bounded
distributive lattices than CRDSA, bounded distributive pseudo-complemented
lattices for example.
