nLab topologically ringed space

A topologically ringed space is a pair $(X,\mathcal{O}_X)$ where $X$ is topological space and $\mathcal{O}_X$ is a sheaf of topological rings. Most often, e.g. in the standard theory of formal schemes, one has a sheaf of complete local rings.

The theory is very similar and generalizes the discrete case of the ringed spaces.