Schemes over topological rings I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ring comes equipped with a given topology.  For example, I am curious to investigate schemes over $C ^\infty (\mathbb{R}^d)$, where it is obvious from the beginning that it would be 'wrong' to ignore the canonical Fréchet space topology (for example, the 'correct' definition of $\mathrm{Spec}\, (R)$ in this case should probably be the collection of all closed prime ideals).  (Part of the motivation for this is, in the spirit of Grothendieck's famous quote, to replace a bad category of only good objects (the smooth category) with a good category containing some bad objects.)
I can't imagine that people haven't investigated things like this before.  On the other hand, I don't know what this subject would be called, and so I don't know where to begin looking to read up on the subject.  Could someone help point me in the right direction?
 A: Although I agree with your statement that it would be wrong to forget the topology of $R$, I would say that there is little evidence to support the idea that the set of closed prime ideals is the `right' definition for Spec(R). This is certainly not the case for analytic geometry over a non-Archimedean field, which has many interesting points not related to prime ideals at all.
In general what you often do is try to define a topos and then identify which objects in the topos are sufficiently `geometric' to deserve your attention. If you really must, you can also try to identify a good family of points. Some nice examples are Spivak's derived manifolds and $C^\infty$-schemes (which are dual to the smooth algebras that Qiaochu mentioned). Both of these are essentially some purely formal extension of the category of manifolds, and do not really involve any understanding of the topological algebra of $C^\infty$ functions.
In a different direction, Alain Connes's flavour of non-commutative geometry really depends on functional analysis of Dirac operators on nuclear Fréchet spaces - but as far as I know, there is no underlying topological space (or topos) in this case. Perhaps the the theory of schemes you are looking for will eventually involve a combination of these ideas.
