The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.

Edit: Due to the comments this type of result seems unlikely. However, I'm tempted to ask what are such criteria on $C^1(X)$ or $C^{\infty}(X)$; assuming that $X$ is a compact sub-manifold with boundary of $\mathbb{R}^K$, for some finite $K$. Here we take the ususal Fréchet topologies on these spaces...

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.

Edit: Due to the comments this type of result seems unlikely. However, I'm tempted to ask what are such criteria on $C^1(X)$ or $C^{\infty}(X)$; assuming that $X$ is a compact sub-manifold with boundary of $\mathbb{R}^K$, for some finite $K$. Here we take the completion of the metric topology $$ d(f,g) := \sum_{n =0}^k \frac1{2^{n+A(n)}} \sum_{|\alpha|\leq n}\frac{ \sup_{x \in X} \frac{\partial}{\partial x^{\alpha}} \|f(x)-g(x)\| }{ 1+ \sup_{x \in X} \frac{\partial}{\partial x^{\alpha}} \|f(x)-g(x)\| } $$ as described by this nice post.

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.

Edit: Due to the comments this type of result seems unlikely. However, I'm tempted to ask what are such criteria on $C^1(X)$ or $C^{\infty}(X)$; assuming that $X$ is a compact sub-manifold with boundary of $\mathbb{R}^K$, for some finite $K$. Here we take the ususal Fréchet topologies on these spaces...

The Stone-Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.

Are any similar results for density in $C_b(X)$ when it is equipped with the topology of uniform convergence? Here, I'm now assuming that $X$ is a Euclidean space.