Nagata's theorem: the topological ring $C_p(X)$ determines $X$

For a Tychonoff space $X$ let $C_p(X)$ be the ring of continuous real functions on $X$ carrying the topology of pointwise convergence, that is, the subspace topology from the product ${\mathbb{R}}^X$.

Nagata's theorem. For Tychonoff $X$ and $Y$, if $C_p(X)$ and $C_p(Y)$ are isomorphic as topological rings, then $X$ and $Y$ are homeomorphic.

Not proved in this library. Recorded with a citation; $C_p$-theory needs both the function space topology and the algebra of $C(X)$ to be developed.

What would prove it. Recover the points of $X$ inside $C_p(X)$ intrinsically: the evaluation maps are exactly the ring homomorphisms $C_p(X)\to \mathbb{R}$ that are continuous for the pointwise topology, and the topology they inherit as a subspace of the dual is the topology of $X$. Continuity of the ring isomorphism is what pins down $X$ rather than $\beta X$.

Why it matters here. It is the sharpest entry in the algebra and topology dictionary, and the comparison with its neighbours is the point. The ring structure alone recovers only $\beta X$ (Gelfand-Kolmogorov theorem: the maximal ideals of $C(X)$ are the points of $\beta X$ ‡); a linear homeomorphism of $C_p(X)$ with $C_p(Y)$, so-called $l$-equivalence, does not recover $X$ at all; but the ring structure together with the topology of pointwise convergence recovers $X$ exactly.