The Stacks project

Example 6.9.3. Let $X$ be a topological space. For each open $U \subset X$ consider the $\mathbf{R}$-algebra $\mathcal{C}^{0}(U) = \{ f : U \to \mathbf{R} \mid f\text{ is continuous}\} $. There are obvious restriction mappings that turn this into a presheaf of $\mathbf{R}$-algebras over $X$. By Example 6.7.3 it is a sheaf of sets. Hence by the Lemma 6.9.2 it is a sheaf of $\mathbf{R}$-algebras over $X$.