In this paper: http://www-math.mit.edu/~hrm/papers/ss.pdf the author claims that Leray originally developed sheaves over closed sets rather than open sets and that it was Cartan who later realized that working with open sets was more natural. Why is this? Are there meaningful/essential differences between both approaches?
Edit:
The bounty goes to the person who can answer all the previous questions and elaborates a little bit more.
I have the feeling that maybe, since Leray was working on functional analysis, he was thinking about subspaces, hence the closure condition.
Many important examples are naturally defined on open sets and not closed sets. For example, consider the sheaf of differentiable functions on a smooth manifold, or the sheaf of holomorphic functions on a complex manifold: these are naturally conditions that you check on open subsets, and it's not even clear what it means to check them on closed subsets.
It also seems to be more natural to talk about open covers rather than closed covers, e.g. the condition that you have a pair of functions on $U$ and $V$ that agree on $U \cap V$ is much stronger if $U, V$ are open (since then $U \cap V$ will also be open) than if $U, V$ are closed. For example, on an $n$-dimensional manifold, an open subset is always also an $n$-dimensional manifold, but a closed subset doesn't even need to be a manifold, and could even be as small as a point. So the condition that you have two sections on $U$ and on $V$ that agree on $U \cap V$ is potentially much weaker.