# Equivalent definitions for fine sheaves

There are some different definitions for fine sheaves.

Let X is a paracompact Hausdorff space, a sheaf F over X is a fine sheaf, if

a) Hom(F,F) is soft

b) For every two disjoint closed subsets A,B$\subset$X, A$\cap$B=$\emptyset$ , there is an endomorphism of the sheaf F$\to$ F which restricts to the identity in a neighborhood of A and to the 0 endomorphism in a neighborhood of B.

c) For every open cover {U$_i$} of X, there is a partition of unity 1= $\sum$ f$_i$ (where the sum is locally finite) subordinate to this covering.

The definition a) appears in the Bredon's book Sheaf theory and you can see the definitions b) and c) at nLab here.

Can you show they are equivalent?

Here is the unanswered same question.

• In Godement's wonderful book "Topologie algebrique et theories des faisceaux", (a) is the definition of fineness (look near top of p. 157). That (a) implies (c) is a special case of Theorem 3.6.1 in Chapter II. To see (c) implies (b), note that $\{X-A, X-B\}$ is an open cover of $X$, so a partition of 1 subordinate to that cover yields (b). Finally, that (b) implies (a) is a special case of Theorem 3.7.2 in Chapter II (applied to the sheaf of rings $\mathscr{H}om(\mathscr{F}, \mathscr{F})$). – nfdc23 May 14 '16 at 14:59
• Thanks, does this Godement's wonderful book has English version, or I have to translate it by computer? – Strongart May 16 '16 at 4:44
• There is no English translation. But mathematical French is not real French. I am bad at foreign languages and used physical copies of French and English versions of a book by Serre to pick up many of the little words (not so many, in reality) which come up in math, and others look similar enough to English that context handles the rest (if one can read English). It is very much worth the effort to acquire some basic ability at reading math French. (I have read thousands of pages of math French and cannot read a French menu, sorry to say. The skills are unrelated.) – nfdc23 May 17 '16 at 2:16
• It is a good way, I can guess some similar enough words, it looks like secret message breaking, haha. – Strongart May 17 '16 at 4:44
• 于老屌，你啃老诈骗，罪恶深重，不配为人！ – jhgfd Nov 8 '18 at 13:12