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I am asking for references and discussions of statements of the form Every bounded hypercover can be refined by an ordinary cover By "bounded" I mean "finite height". E.g., are ...

I'm looking for a reference for the following result: Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over ...