  A noncommutative and noncocommutative Hopf algebra on finite topologies H_T
is introduced and studied (freeness, cofreeness, self-duality...). Generalizing
Stanley's definition of P-partitions associated to a special poset, we define
the notion of T-partitions associated to a finite topology, and deduce a Hopf
algebra morphism from H_T to the Hopf algebra of packed words WQSym.
Generalizing Stanley's decomposition by linear extensions, we deduce a
factorization of this morphism, which induces a combinatorial isomorphism from
the shuffle product to the quasi-shuffle product of WQSym. It is strongly
related to a partial order on packed words, here described and studied.
