  A topological commutative ring is said to be rigid when for every set $X$,
the topological dual of the $X$-fold topological product of the ring is
isomorphic to the free module over $X$. Examples are fields with a ring
topology, discrete rings, and normed algebras. Rigidity translates into a dual
equivalence between categories of free modules and of "topologically-free"
modules and, with a suitable topological tensor product for the latter, one
proves that it lifts to an equivalence between monoids in this category (some
suitably generalized topological algebras) and coalgebras. In particular, we
provide a description of its relationship with the standard duality between
algebras and coalgebras, namely finite duality.

Commutative Algebra;Category Theory;General Topology