Unital, point-separating, and nowhere-vanishing real function algebras on a compact Hausdorff space

Let $X$ be a compact Hausdorff space (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). A subset $A\subseteq C(X,\mathbb{R})$ is a real function algebra when it is a real vector subspace under the pointwise operations of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$ and is closed under the pointwise multiplication of The ring $R^X$ of all functions from a set $X$ into a ring, with pointwise operations. Every member is continuous in the sense of Continuity of a map of topological spaces at a point and globally.

The algebra $A$ is:

• unital when it contains every constant real-valued function;
• point-separating when for every distinct $x,y\in X$ there is $f\in A$ with $f(x)\mathrm{\ne }f(y)$;
• nowhere-vanishing when for every $x\in X$ there is $f\in A$ with $f(x)\mathrm{\ne }0$.

Unitality implies nowhere-vanishing when $X$ is nonempty, but nowhere-vanishing does not assume that the constant-one function belongs to $A$.

Uniform approximation on this page. For $F\subseteq C(X,\mathbb{R})$ and $f\in C(X,\mathbb{R})$, $f$ is uniformly approximable by members of $F$ means that for every $\epsilon >0$ there is $g\in F$ with $\mathrm{\mid }f(x)-g(x)\mathrm{\mid }<\epsilon$ for every $x\in X$; the uniform closure of $F$ is the set of members of $C(X,\mathbb{R})$ uniformly approximable by members of $F$, and $F$ is uniformly dense when that closure is all of $C(X,\mathbb{R})$. Stated this way the notion is available for every $X$, the empty space included. For nonempty $X$ it is exactly density for the topology of uniform convergence of Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on $Y^X$ and on $C(X,Y)$, whose uniform metric is defined only on a nonempty domain.