  This is both an expository and research paper where we advocate a systematic
study of continuous analogues of finite partially ordered sets, convex
polytopes, oriented matroids, arrangements of subspaces, finite simplicial
complexes, and other combinatorial structures. Among the illustrative examples
are an Euler formula for a class of `continuous convex polytopes' (conjectured
by Kalai and Wigderson), a duality result for a class of `continuous matroids',
a calculation of the Euler characteristic of ideals in the Grassmannian poset
(related to a problem of Gian-Carlo Rota), an exposition of the `homotopy
complementation formula' for topological posets and its relation to the results
of Kallel and Karoui about `weighted barycenter spaces' and a conjecture of
Vassiliev about simplicial resolutions of singularities. We also include an
extension of the index inequality (Sarkaria's inequality) based on interpreting
diagrams of spaces as continuous posets.
