# Which completely regular Hausdorff spaces admit a proper map to $\mathbb R$?

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?

If "proper" means "the preimages of compact sets are compact", then the class of Tychonoff spaces admitting proper maps to $\mathbb R$ coincides with the class of $\sigma$-compact locally compact regular spaces (so, is well-known and well-studied).