Unital point-separating real vector sublattices of $C(X,\mathbb{R})$

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 $L\subseteq C(X,\mathbb{R})$ is a real vector sublattice 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, for every $f,g\in L$, it contains the pointwise functions $f\vee g:x⟼\max \{f(x),g(x)\},f\wedge g:x⟼\min \{f(x),g(x)\}\mathrm{.}$

The vector sublattice $L$ is unital when it contains every constant real-valued function, and it separates points when for every distinct $x,y\in X$ there is $g\in L$ with $g(x)\mathrm{\ne }g(y)$. Every member of $L$ is continuous in the sense of Continuity of a map of topological spaces at a point and globally.

The vector-space hypothesis is part of the definition used here. Closure under pointwise maxima and minima alone does not supply the affine rescaling required for two-point interpolation.