Topological characterization of various types of $\mathcal{C}^{\infty}$-rings

Topologies on algebraic and equational theories are used to define germ determined, near-point determined and point determined $\mathcal{C}^{\infty}$-rings, without requiring them to be finitely generated. It is proved that any $\mathbb{R}$-algebra morphism (without requiring continuity) into a near-point determined $\mathcal{C}^{\infty}$ring is a $\mathcal{C}^{\infty}$ morphism (and hence continuous).

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Published 17 December 2014