  The aim of this paper is sketch a theory of divisibility and factorisation in
topological monoids, where finite products are replaced by convergent products.
The algebraic case can then be viewed as the special case of discretely
topologised topological monoids.
  In particular, we define the topological factorisation monoid, a
generalisation of the factorisation monoid for algebraic monoids, and show that
it is always topologically factorial: any element can be uniquely written as a
convergent product of irreducible elements. We give some sufficient conditions
for a topological monoid to be topologically factorial.
