Locally topologically ringed space

Definition

A locally topologically ringed space is a pair $(X,\mathcal O_X)$ consisting of a topological space $X$ and a sheaf $\mathcal O_X$ of commutative topological rings such that:

every restriction map $\mathcal O_X(U)\to\mathcal O_X(V)$ , for $V\subseteq U$ , is continuous;
after forgetting the topologies on the rings, $(X,\mathcal O_X)$ is a locally ringed space.

In the adic formal-scheme setting, the topologies on local sections are linear topologies defined by ideals, and the structure sheaf is complete for those topologies.

Morphisms

A morphism

(f,f^\#):(X,\mathcal O_X)\longrightarrow(Y,\mathcal O_Y)

consists of a continuous map $f:X\to Y$ and a morphism of structure sheaves

f^\#:\mathcal O_Y\longrightarrow f_*\mathcal O_X

whose maps on sections are continuous and whose induced homomorphisms on stalks are local. Formal schemes use morphisms of this kind.

References

The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY.