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Let $X$ be some space, I have basically 2 questions:

1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on paracompact spaces are acyclic, but a lot of people require for paracompact spaces to be Hausdorff too, so I don't know if this works on non-Hausdorff paracompact spaces, and...

2 - In case soft sheaves are acyclic on non-Hausdorff paracompact spaces. Why don't I define the trivial topology on $X$? (Spaces with the trivial topology are paracompact) That way aren't all the sections in the sheaf trivially global sections? I know it's kind of dumb but wouldn't that automatically make the sheaves soft?

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  • $\begingroup$ Richard -- what is a paracompact but non-Hausdorff space? $\endgroup$
    – algori
    Commented Feb 5, 2012 at 2:47
  • $\begingroup$ Well I was doing some reading and a space with the trivial topology is not Hausdorff, but is paracompact, hence the question, I read that some people require that paracompact spaces be Hausdorff, but some don't, so what gives? @algori $\endgroup$
    – Richard Jennings
    Commented Feb 5, 2012 at 2:57
  • $\begingroup$ Richard -- the Hausdorff condition is usually part of the definition of a paracompact space. If you want to use a different version, that's fine, but then could you please specify the definition you are using. $\endgroup$
    – algori
    Commented Feb 5, 2012 at 3:11
  • $\begingroup$ Here's a shot at question 2, assuming by "trivial topology" you mean the indiscrete topology. In that case, you only have one (non-empty) open set, so you're right that all sections are global sections. But that means that a sheaf is no different from a group (which you assign to the non-empty open set). In this way, the sheaf theory is just the same as the sheaf theory over a point. And then indeed every sheaf is soft and acyclic. But I don't see how this would relate back to every other non-Hausdorff paracompact space, whatever we all decide they should be. $\endgroup$
    – Greg Friedman
    Commented Feb 5, 2012 at 3:32
  • $\begingroup$ @Greg Friedman Can you elaborate more on what you're saying? I'm studying up on this and still learning, thanks $\endgroup$
    – Richard Jennings
    Commented Mar 2, 2012 at 20:18

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