Nagata Cp theorem remains topological

For Tychonoff spaces $X$ and $Y$, an isomorphism of topological rings $C_p(X,\mathbb{R})\cong C_p(Y,\mathbb{R})$, where $C_p$ carries the topology of pointwise convergence inherited from ${\mathbb{R}}^X$ and ${\mathbb{R}}^Y$, implies $X\cong Y$.

This result is recorded, not proved here, and it is distinct from the ring-only description of the Stone–Čech compactification: the local Gelfand–Kolmogorov theorem identifies the maximal ideals of $C(X,\mathbb{R})$ set-theoretically with the points of $\beta X$ and detects which of those ideals are fixed (Gelfand-Kolmogorov for rings of continuous functions), but does not assert that the ring alone reconstructs the topology of $\beta X$. Nagata's theorem instead includes the pointwise topology as part of the data and recovers $X$ itself. The catalogue target is Nagata's theorem: the topological ring $C_p(X)$ determines $X$ ‡, and it stays a Recorded result.

• Orientation only. No item on this page depends on this statement; it is not a supplier, and it is excluded from every proof path.
• Open obligation for the catalogue. Full-text verification of the exact statement, hypotheses and the pointwise-topology convention is an open repair obligation on the catalogue target, not a fact established here.