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This is a follow up to this question of mine.

First of all, let me fix some terminologies, which may or may not be standard:

Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)$ where $|X|$ is a topological space and $\mathcal{O}_X$ is a sheaf of (commutative) topological rings. Such a space is said to be a topological locally ringed space if the stalks of the structure sheaf $\mathcal{O}_X$ are all local rings.

My questions are then:

  1. Are there issues with these definitions of mine ?
  2. Is there a decently extensive reference on these kinds of topological ringed spaces ? The only thing I have managed to find is this tiny nLab article.

Thank you!

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    $\begingroup$ Your definition is exactly the same as the one in The Stacks Project and in EGA! The former has a small bit of material on locally topologically ringed spaces (LTRSs) in Tag 0AHY, while the later defines those in EGA 0, Section 4 and talks a bit more about them in the section on formal schemes (EGA 1, Section 10). Both the Stacks Project and EGA pass very quickly to formal schemes, though, and I also don't know of an extensive general reference for LTRSs :/ $\endgroup$
    – Emily
    Commented Sep 2, 2021 at 4:36

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