The space of ideals of C^\infty(M,R)

TL;DR

This paper introduces a topology on the set of codimension d ideals in the algebra of smooth functions on a compact manifold, creating a compact Hausdorff space that compactifies the configuration space of points in the manifold.

Contribution

It defines a new topology on ideals of C^ abla(M,R) that includes the configuration space and has a rich, semialgebraic structure over the symmetric product.

Findings

The space is compact Hausdorff.

Contains the configuration space as a subspace.

Has a natural semialgebraic chart structure.

Abstract

The paper is concerned with defining a topology on the set of ideals of codimension d of the algebra C^\infty(M,R) with M being a compact smooth manifold. Its main property is that it is compact Hausdorff and it contains as a subspace the configuration space of d distinct unordered points in M and therefore provides a "compactification" of this configuration space. It naturally forms a space over the symmetric product SP_d(M) and as such has a very rich structure. Moreover it is covered by a (naturally defined) set of charts in which it is semialgebraic.