= Ringed space
{title2=$(X,\mathcal O_X)$}
{wiki}
A <ringed space> $(X,\mathcal O_X)$ is a <topological space> together with a <sheaf of rings> describing its local <functions>. For an open subset $U$, $\mathcal O_X(U)$ is the <ring> of sections over $U$, with compatible restrictions and gluing. A <smooth manifold> with its real-valued smooth <functions> is a <ringed space>; a <complex manifold> with holomorphic <functions> is another. The underlying <topology> alone does not distinguish these kinds of <functions>. A <locally ringed space> additionally requires every <stalk> to be a <local ring>, which is the condition used in the definition of a <scheme>.
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