Continuous real-valued functions make a space into a locally ringed space

The usual restriction of continuous functions makes $U\mapsto {\mathcal{C}}_X^0(U)$ a sheaf of commutative rings on $X$, so [F1] gives a ringed space $(X,{\mathcal{C}}_X^0)$.

Fix $x\in X$. By [F2], the stalk ${\mathcal{C}}_{X,x}^0$ consists of germs of continuous real-valued functions near $x$. The germs vanishing at $x$ form an ideal ${\mathfrak{m}}_x$. If a germ is not in ${\mathfrak{m}}_x$, it has a representative $g$ with $g(x)\mathrm{\ne }0$, so continuity makes $g$ nonzero on a smaller neighbourhood of $x$; hence $1\mathrm{/}g$ is continuous there and defines an inverse germ. Therefore the nonunits are exactly the germs in ${\mathfrak{m}}_x$, so ${\mathfrak{m}}_x$ is the unique maximal ideal. Thus ${\mathcal{C}}_{X,x}^0$ is a local ring, and [L1] shows that $(X,{\mathcal{C}}_X^0)$ is locally ringed.