  We remark that Pearl's Graphoid intersection property, also called
intersection property in Bayesian networks, is a particular case of a general
intersection property, in the sense of intersection of coverings, for
factorisation spaces, also coined as factorisation models, factor graphs or by
Lauritzen in his reference book 'Graphical Models' as hierarchical model
subspaces. A particular case of this intersection property appears in
Lauritzen's book as a consequence of the decomposition into interaction
subspaces; the novel proof that we give of this result allows us to extend it
in the most general setting. It also allows us to give a direct and new proof
of the Hammersley-Clifford theorem transposing and reducing it to a
corresponding statement for graphs, justifying formally the geometric intuition
of independency, and extending it to non finite graphs. This intersection
property is the starting point for a generalization of the decomposition into
interaction subspaces to collections of vector spaces.
