Characters on algebras of smooth functions. (English) Zbl 0691.58020

A smooth space is a pair (M,\({\mathcal S})\) where M is a set and \({\mathcal S}^ a \)collection of real-valued functions on M which separates points, is closed under composition with \(C^{\infty}\)-functions and is closed under locally finite (in an obvious sense) sums. The smooth space (M,\({\mathcal S})\) is smoothly real-compact provided that any algebraic homomorphism \({\mathcal S}\to {\mathbb{R}}\) is an evaluation at a point of M. It is shown that if (M,\({\mathcal S})\) is a smoothly real-compact smooth space then the initial topology on M induced by \({\mathcal S}\) is real-compact. Provided \({\mathcal S}\) is dense in the set of all continuous functions in the topology of uniform convergence then the converse of this statement is also true.
Reviewer: D.B.Gauld


58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
54D60 Realcompactness and realcompactification
58A05 Differentiable manifolds, foundations
58C05 Real-valued functions on manifolds
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