Since $\mathbb R$ is a topological ring, the representable contravariant functor $\mathrm{Hom}_{Top}(-,\mathbb R)$ sends topological spaces to (unital, commutative and associative) $\mathbb R$-algebras. Consequently, it determines a covariant functor from $\mathrm{Top}$ to affine $\mathbb R$-schemes.
It is a straightforward fact that $\mathrm{Hom}_{Top}(-,\mathbb R)$ is faithful, i.e. injective, on morphisms with codomain $X$ if and only if $X$ is completely Hausdorff, i.e. if and only if for any pair of points $x$ and $y$ there is a function $f\in\mathrm{Hom}_{Top}(-,\mathbb R)$ such that $f(x)\neq f(y)$. In other words, the category of completely Hausdorff spaces is a subcategory of the category of affine $\mathbb R$-schemes.
It is also straightforward to check that, when $X$ is completely regular, i.e. for any closed subset $K\subseteq X$ and $x\not\in K$ there exists a function $f\in\mathrm{Hom}_{Top}(X,\mathbb R)$ such that $f(K)=0$ and $f(x)\neq0$, then $\mathrm{Hom}_{Top}(-,\mathbb R)$ is full, i.e. surjective, on morphisms with codomain $X$ if and only if it is full on morphisms $\{*\}\to X$.
In other words, complete regularity of $X$ reduces the problem of fullness of $\mathrm{Hom}_{Top}(-,\mathbb R)$ on morphisms with codomain $X$ to the problem of whether every $\mathbb R$-algebra homomorphism $\mathrm{Hom}_{Top}(X,\mathbb R)\to\mathbb R$ is given by evaluation at a point $x\in X$. Furthermore, complete regularity implies that the topology on $X$ is the topology induced on kernels of evaluation homomorphisms $\mathrm{Hom}_{Top}(X,\mathbb R)\xrightarrow{\mathrm{ev}_x}\mathbb R$ by the Zariski topology of $\mathrm{Hom}_{Top}(X,\mathbb R)$.
It turns out that a sufficient condition for every $\mathbb R$-algebra homomorphism $\mathrm{Hom}_{Top}(X,\mathbb R)\to\mathbb R$ to be an evaluation homomorphism is that there exists a function $f\in\mathrm{Hom}_{Top}(X,\mathbb R)$ such that each fiber $f^{-1}(r)\subseteq X$ for $r\in\mathbb R$ is compact, i.e. that there exists a proper map $X\xrightarrow{f}\mathbb R$. These of course include compact Hausdorff spaces, but also second-countable topological manifolds (using their closed embeddings in $\mathbb R^n$ and the square-distance function from the origin), thereby establishing the categories of compact Hausdorff spaces and the category of topological manifolds as full subcategories of the category of affine $\mathbb R$-schemes. In fact the above argument goes through for smooth manifolds (because a function between smooth manifolds is smooth if and only if pre-composition with it sends smooth functions to smooth functions).
Have completely regular (Hausdorff) topological spaces $X$ admitting a proper map $X\xrightarrow{f}\mathbb R$ been studied? Is there some kind of classification, or some class of spaces beyond manifolds and compact Hausdorff spaces that have the property?
