In Récoltes et Semailles, Grothendieck explains he invented the concept of scheme to unify algebraic varieties on algebraically closed fields, arithmetical varieties in characteristic $p$, and also the usual metric spaces. This last part is not obvious to me: if we start with a metric space $(X,d)$ we do have a topological space $X$, but we need a structure sheaf of rings to replace the metric distance $d$ (so that we can distinguish homeomorphic spaces with different metric structure, like a triangle and a circle). The sheaf of continuous functions $\mathcal{C}(X,\mathbb{R})$ seems a natural choice, because distances are $\mathbb{R}$-valued, and because its stalks are local rings. Did Grothendieck have this sheaf in mind to represent metric spaces as schemes? Does this sheaf distinguish homeomorphic spaces with different metric structures?
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3I don't think $\mathcal C(X, \mathbb R)$ can distinguish a triangle from a circle. If you have a homeomorphism $f: S^1 \to T$ (where $T$ is a triangle), then any continuous function $T \to \mathbb R$ gives a continuous function $S^1 \to \mathbb R$ by composition, and vice versa. Put differently, the sheaf $\mathcal C(X, \mathbb R)$ knows nothing about the metric, but depends only on the topology. – red_trumpet Feb 15 '22 at 12:04
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@red_trumpet Indeed. Do you see another sheaf to represent the metric structure? – V. Semeria Feb 15 '22 at 21:26