An order-theoretic notion of convergence tailored to a Riesz ideal I of the continuous real-valued functions on a Lindelof locally compact Hausdorff space: a sequence converges strictly when it is dominated by a decreasing sequence with pointwise infimum zero. This convergence is equivalent to uniform convergence on compacts together with a single order bound inside I, and it makes the closure of a strictly point-separating, nowhere-vanishing Riesz subspace (or bounded subalgebra) equal to all of I - a Stone-Weierstrass theorem valid on ideals that are not closed in the Frechet topology. Students learn how such density arguments force two strictly continuous positive linear maps that agree on a point-separating set to agree everywhere, and how that yields determinacy of moment problems without spectral theory.
This paper studies lattice-theoretic properties of spaces of continuous functions, focusing on Riesz ideals within the ordered vector space structure of continuous function spaces. It examines Stone-…