Additional structures on $C_p(X)$

With the operations of pointwise addition and pointwise multiplication, $C_p(X)$ is a commutative topological ring with unit, the unit being the constant function with value 1. It is a famous theorem of Gel$'$fand and Kolmogoroff [18] that the ring structure by itself determines the topological structure on $X$ provided $X$ is compact. They proved that if $X$ and $Y$ are compact and $C(X)$ and $C(Y)$ are isomorphic as rings then $X$ and $Y$ are homeomorphic. For details, see also Dugundji [17, Theorem XIII.6.5]. (The proof in [17] makes use of the topology of pointwise convergence.)

For noncompact spaces $X$, the algebraic structure of $C(X)$ is, in general, not strong enough to determine the topology of $X$. For consider the spaces $X=\omega_1$ and $Y=\omega_1+1$. Then clearly $C(X)$ and $C(Y)$ are isomorphic as rings, but $X$ and $Y$ are not homeomorphic.

For arbitrary spaces there is a result in the same spirit though. Nagata [27]proved that $C_p(X)$ and $C_p(Y)$ are topologically isomorphic as topological rings if and only if $X$ and $Y$ are homeomorphic. That we deal with real valued functions is essential in this result. It was shown in Arhangel$'$ski [3, page 12] that the ring of all continuous functions $X\to{\mathbb R}^\infty$, endowed with the topology of pointwise convergence, does not always determine the topological type of $X$.