Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by $\widetilde{f}(x)=f(x)$ if $x\in U$ and $\widetilde{f}(x)=0$ otherwise.
Let $G(M)$ be the set of all functions on $M$ which are finite linear combinations of functions of the form $\widetilde{f}$ for various $f$s and various $U$s.
Note that, in the case where $M$ is Hausdorff we have that $G(M)$ is precisely the set of compactly-supported smooth functions on $M$. However when $M$ is not Hausdorff, $G(M)$ contains non-continuous functions.
My question is, has this class of function $G(M)$ appeared in the literature before? Does it have a standard name?
