subsheaf of abelian groups

Let $ℱ$ be a sheaf of abelian groups over a topological space $X$. Let $𝒢$ be a sheafover $X$, such that for every open set $U\subset X$, $𝒢(U)$ is a subgroup of$ℱ(U)$. And further let the on $𝒢$ be by those on $ℱ$.Then $𝒢$ is a subsheaf of $ℱ$.

Suppose a sheaf of abelian groups $ℱ$ is defined as a disjoint union of stalks ${ℱ}_x$ over points $x\in X$, and $ℱ$ is topologized in the appropriate manner.In particular, each stalk is an abelian group and the group operations are continuous.Then a subsheaf $𝒢$ is an open subset of $ℱ$ such that ${𝒢}_x=𝒢\cap {ℱ}_x$ is a subgroup of ${ℱ}_x$.

When $𝒢$ is a subsheaf of $ℱ$, then${ℱ}_x/{𝒢}_x$ is an abelian group. Considering this to be the stalk over $x$we have a sheaf which is denoted by $ℱ/𝒢$, with the topology being the quotient topology.

Example.

Suppose $M$ is a complex manifold.Let ${\mathrm{M}}^{\mathrm{*}}$ be the sheaf of germs of meromorphic functions which are not identically zero. That is, for $z\mathrm{\in }M\mathrm{,}$ the stalk ${\mathrm{M}}_z^{\mathrm{*}}$ is the abelian group of germs of meromorphic functions at $z$ with the group operation being multiplication.Then ${\mathrm{O}}^{\mathrm{*}}$, the sheafof germs of holomorphic functions which are not identically 0 is a subsheafof ${\mathrm{M}}^{\mathrm{*}}$.

The sheaf ${\mathrm{M}}^{\mathrm{*}}\mathrm{/}{\mathrm{O}}^{\mathrm{*}}$ is then the sheaf of divisors. If $M$ is of (complex) dimension 1, then${\mathrm{M}}^{\mathrm{*}}\mathrm{/}{\mathrm{O}}^{\mathrm{*}}$ is just the sheaf of functions into the integers with finite support.