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:
- Are there issues with these definitions of mine ?
- 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!
