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replaced deprecated tag 'topology'; added relevant tags
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Ricardo Andrade
Ricardo Andrade
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Added clarification about the pathological nature of these spaces.
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anon
anon
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The question is in the title. A more precise formulation is:

Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?

The obvious example is a discrete space. I'd be happy with a characterization of compact Hausdorff topological spaces $X$ satisfying the above property.

Edit: Following Georges Elencwajg's answer, I would like to clarify that these spaces will be quite pathological from the viewpoint of classical topology. Nevertheless, I do not know a single example which does satisfy the above vanishing property and is not discrete. For example, does the Cantor set have this property?

The question is in the title. A more precise formulation is:

Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?

The obvious example is a discrete space. I'd be happy with a characterization of compact Hausdorff topological spaces $X$ satisfying the above property.

The question is in the title. A more precise formulation is:

Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?

The obvious example is a discrete space. I'd be happy with a characterization of compact Hausdorff topological spaces $X$ satisfying the above property.

Edit: Following Georges Elencwajg's answer, I would like to clarify that these spaces will be quite pathological from the viewpoint of classical topology. Nevertheless, I do not know a single example which does satisfy the above vanishing property and is not discrete. For example, does the Cantor set have this property?

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anon
anon
  • 281
  • 2
  • 7

When does the sheaf cohomology of a topological space vanish?

The question is in the title. A more precise formulation is:

Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?

The obvious example is a discrete space. I'd be happy with a characterization of compact Hausdorff topological spaces $X$ satisfying the above property.