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NIce, thanks!
Never thought about that, but it makes sense – thanks!
Strangely enough there is an empty space missing in the first line of empty space as it says Theempty space. Fun!
Tim, I saw that too, but didn’t see how to fix it.
I have fixed it for now by removing the link but this is another bug to tell Richard.
after the lines
that the logarithm of exponential generating functions of some type of objects should be the exponential generating function for the connected objects of that type. Since this logarithm has no constant term, this suggests the empty object should not count as connected. This result is also known in the physics literature as the linked-cluster theorem
I added the pointer:
(see this Prop. at geometry of physics – perturbative quantum field theory).
I do not understand the counterpoint about the uniqueness of connected component decomposition failing when the empty set is connected. The decomposition is defined to be into maximal (path-)connected components in order to be unique. Since the empty set is never maximal, it would never appear in such a decomposition. It would be like saying the decomposition is non-unique since $[0,2] = [0,1) \cup [1,2]$.
Re #8, the decomposition you refer to is meant to be a decomposition into a coproduct; i.e. the $\cup$ is meant to be a disjoint union. So $[0,1) \cup [1,2] = [0,2]$ is not the sort of decomposition it alludes to.
I have changed the “$\cup$“s to “$\sqcup$“s (here).
Maybe it was technically correct as it was, given that the ambient text suggests that the spaces in the union are assumed to be disjoint. Or maybe it’s not so clear, since the text is discussing a counter-factual property of the empty set.
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