Suppose I have a topological space $X$ and a cover $S_i$ of $X$. We say $X$ is coherent with respect to the cover $S_i$ if $X$ has the weak topology with respect to $S_i$, that is, if $S \subset X$ is open iff $S \cap S_i$ is open for each $i$.
A nice fact is that if $X$ is coherent with respect to $S_i$ then $f:X \to Y$ is continuous iff its restriction to each $S_i$ is continuous, or in other words, in $\mathsf{Top}$,
$$f:X \to Y \text{is a morphism iff its restriction to each $S_i$ is a morphism.} \tag{1}$$ The main facts I keep in mind are that
- An open cover is coherent (or more generally if the interiors of $S_i$ form a cover, then $S_i$ is coherent)
- A locally finite closed cover is coherent (or more generally, if the closures of a cover $S_i$ are locally finite, then $S_i$ is coherent).
My question is
What types of covers have property (1) in the category of smooth manifolds?
What kind of covers can we expect to have property (1)? Certainly open covers still work.
I believe this condition is tied up in the notion of a sheaf?