View source for Soft sheaf

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A [[Sheaf|sheaf]] of sets $\mathcal F$ on a topological space $X$ any section of which over some closed subset in $X$ can be extended to a section of $\mathcal F$ over all of $X$. Examples of soft sheaves are: the sheaf of germs of discontinuous sections of an arbitrary sheaf of sets on $X$; any [[Flabby sheaf|flabby sheaf]] $\mathcal F$ on a [[Paracompact space|paracompact space]] $X$; and any [[Fine sheaf|fine sheaf]] $\mathcal F$ of Abelian groups on a paracompact space $X$. The property of softness of a sheaf $\mathcal F$ on a paracompact space $X$ is local: A sheaf $\mathcal F$ is soft if and only if any $x\in X$ has an open neighbourhood $U$ such that $\mathcal F|_U$ is a soft sheaf on $U$. A soft sheaf on a paracompact space induces a soft sheaf on any closed (and, if $X$ is metrizable, any locally closed) subspace. A sheaf of modules over a soft sheaf of rings is a soft sheaf.

If

$$0\to\mathcal F^0\to\mathcal F^1\to\dots$$

is an exact sequence of soft sheaves of Abelian groups on a paracompact space $X$, then the corresponding sequence of groups of sections

$$0\to\mathcal F^0(X)\to\mathcal F^1(X)\to\dots$$

is also exact. The cohomology group $H^p(X,\mathcal F)$ of any soft sheaf $\mathcal F$ of Abelian groups on a paracompact space $X$ is trivial for $p>0$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Bredon,   "Sheaf theory" , McGraw-Hill  (1967)  pp. §9</TD></TR></table>