Let $n\geq 1$ and $F$ be a sheaf on $\mathbb{R}^n$ of continuous functions.
Assume for every open subsets $U,V$ of $\mathbb{R}^n$ and every $f\in F(U)$, $g\in F(V)$, $g\circ f\in F(U\cap f^{-1}(V))$.
$\bullet$ Is there a name to qualify such a sheaf ?
Let $X$ is a set, $(U_i)_{i\in I}$ a collection of non empty subsets covering $X$ and for $i\in I$, let $z_i:U_i\to\mathbb{R}^n$ be and injective function with an open image. If we assume that for every $(i,j)\in I^2$, $z_i(U_i\cap U_j)$ is open and $z_j\circ z_i^{-1}\in F(z_i(U_i\cap U_j))$ then there is a unique topology on X such that each $z_i$ is a homeomorphism onto it's image.
$\bullet$ Is there a name to define such structure on $X$ induced by $F$ ? A $F$-manifold or something like that ?
For example if $n=2$ and $F$ is the sheaf of holomorphic maps on $\mathbb{C}$ then a Riemann surface is any set $X$ with a Hausdorff connected topology induced by $F$. Smooth manifolds are defined is a similar way.