  We study the basic properties of a dual "spectral" topology on positive type
spaces of h-inductive theories and its essential connection to infinitary
logic. The topology is Hausdorff, has the Baire property, and its compactness
characterises positive model completeness; it also has a basis of clopen sets
and is described by the formulas of geometric logic. The "geometric types" are
closed in the type spaces under all the operations of infinitary logic, and we
introduce a positive analogue of existentially universal structures, through
which we interpret the full first order logic in positive type spaces. This
shows how "positive $\omega$-saturation" is a fundamental connection between
positive and infinitary logic, and we suggest a geometric analogue of positive
Morleyisation.
