  In formal concept analysis, the collection of protoconcepts of any context
forms a double Boolean algebra (dBa) which is fully contextual. Semiconcepts of
a context form a pure dBa. The present article is a study on topological
representation results for dBas, and in particular, those for fully contextual
and pure dBas. In this work, a context on topological spaces (CTS) is
considered, and the focus is on a special kind of CTS in which the relation
defining the context as well as the converse of the relation are continuous
with respect to the topologies. Such CTS are denoted as CTSCR. It is observed
that clopen object oriented protoconcepts of a CTSCR form a fully contextual
dBa, while clopen object oriented semiconcepts form a pure dBa. Every dBa is
shown to be quasi-embeddable into the dBa of clopen object oriented
protoconcepts of a particular CTSCR. The quasi-embedding turns into an
embedding in case of a contextual dBa, and into an isomorphism, when the dBa is
fully contextual. For pure dBas, one obtains an isomorphism with the algebra of
clopen object oriented semiconcepts of the CTSCR. Representation of finite dBas
and Boolean algebras are also addressed in the process. Abstraction of
properties of this CTSCR leads to the definition of Stone contexts. Stone
contexts and CTSCR- homeomorphisms are seen to form a category, denoted Scxt.
Furthermore, correspondences are observed between dBa isomorphisms and
CTSCR-homeomorphisms. This motivates a study of categorical duality of dBas,
constituting the second part of the article. Pure dBas and fully contextual
dBas along with dBa isomorphisms form categories, denoted PDBA and FCDBA
respectively. It is established that PDBA is equivalent to FCDBA, while FCDBA
and PDBA are dually equivalent to Scxt.
