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For the usual additive Čech cohomology groups, $\mathcal{F}$ is understood to be a sheaf of abelian groups. Order the index set of the open cover $\mathfrak{U}=(U_i)$. The Čech cochain group isAn empty intersection contributes the zero group. The Čech coboundary is the alternating sum of restriction maps:Every double omission appears twice with opposite sign, so ${\delta }^2=0$. The Čech cohomology is the cohomology of this cochain complex:Here ${\check{C}}^{-1}=0$. The sheaf gluing axiom identifies ${\check{H}}^0(\mathfrak{U},\mathcal{F})$ with $\Gamma (X,\mathcal{F})$. This definition is for the fixed cover; no limit over refinements is part of the group requested here. Without the abelian-group structure, this additive Čech cochain complex is not defined.

Treat $0$ as the trivial group if $G$ is not abelian. The restriction maps satisfy the presheaf identities. To verify the sheaf gluing axiom, take an open cover $(V_i)$ of $U$. If $x\notin U$, every section group is trivial. If $x\in U$, at least one $V_i$ contains $x$. Every two such opens have an overlap containing $x$, so compatibility forces all their sections to have the same value $g\in G$. Opens not containing $x$ have their unique trivial section. The value $g\in \mathcal{G}(U)$ is exactly the unique glued section. Thus the stated presheaf is a skyscraper sheaf, including for nonabelian groups.

For the stalk of a sheaf, take the direct limit over neighbourhoods of $y$. If some neighbourhood $V$ of $y$ omits $x$, neighbourhoods inside $V$ are cofinal and all their groups are trivial. If every neighbourhood of $y$ contains $x$, every group in the system is $G$ and every transition map is the identity. Consequently, on the arbitrary topological space in the question,Membership in this closure means exactly that every neighbourhood of $y$ contains $x$. In particular ${\mathcal{G}}_x=G$. If $x$ is a closed point, the closure is just $\{x\}$ and the familiar point-supported answer results. The separation assumption is not present in the printed question, so it cannot be imposed silently.

We prove the vanishing by an injective resolution of sheaves and the flasque-kernel section-lifting lemma.

For a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ with flasque kernel, every section of $\mathcal{H}$ on an open set lifts to $\mathcal{G}$ there. Indeed choose a maximal partial lift by the Zorn lemma, using the sheaf gluing axiom on chains. At any point outside its domain choose a local lift. The difference on the overlap is a section of $\mathcal{F}$ and extends to the new open set by flasqueness; subtract that extension from the new lift and glue. Maximality forces the lift's domain to be the whole open set. Thus taking sections preserves this short exact sequence.

Every injective sheaf is flasque: for $U\subseteq V$, the monomorphism between the extension by zero sheaves ${\mathbb{Z}}_U\hookrightarrow {\mathbb{Z}}_V$ and the defining extension property of injectivity make the restriction $\mathcal{J}(V)\to \mathcal{J}(U)$ surjective. Embed $\mathcal{F}$ into an injective sheaf $\mathcal{J}$. The quotient $\mathcal{Q}$ is flasque as well: lift a section of $\mathcal{Q}$ on a smaller open to $\mathcal{J}$, extend it in $\mathcal{J}$, and project.

The long exact sequence in sheaf cohomology now gives $H^1(X,\mathcal{F})=0$, since $\Gamma (X,\mathcal{J})\to \Gamma (X,\mathcal{Q})$ is surjective, and for $i\ge 2$ gives $H^i(X,\mathcal{F})\cong H^{i-1}(X,\mathcal{Q})$, since injectives have zero positive-degree derived functors. Repeat the construction for the flasque quotient. After finitely many such shifts any specified positive degree reduces to a vanishing first group. HenceNo separation, compactness or scheme hypothesis is used.

Put $\mathcal{Q}={\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, the quotient in sheaves of abelian groups, where ${\mathcal{K}}_X^{\ast }$ is the sheaf of nonzero rational functions on an irreducible variety and ${\mathcal{O}}_X^{\ast }$ is the sheaf of units of the structure sheaf. At a closed point, the discrete valuation induces an isomorphismThe kernel consists precisely of the units of the discrete valuation ring, and surjectivity follows by taking powers of a uniformizer.

A global section of $\mathcal{Q}$ is locally represented by nonzero rational functions. Since $X$ has a finite affine cover and is Noetherian, it is quasi-compact, so finitely many such representatives suffice. Each has only finitely many zeros and poles; consequently the valuations of the section define a finite divisor on an algebraic curve. A section with zero valuation at every closed point is zero in every stalk and therefore zero. This gives an injective mapIt is surjective as well. For $D={\sum }_P{n}_{P}P$, take a rational uniformizer $t_P$ at each point of its finite support. On a sufficiently small neighborhood of $P$, the function $t_P^{n_P}$ has the prescribed divisor there: remove the other points in its zero and pole set and the other support points of $D$. On the complement of the support use the rational function $1$. On overlaps these representatives have the same valuations at every point, so their ratios are units everywhere on the overlap, and their classes in $\mathcal{Q}$ agree. The sheaf gluing axiom now supplies a global section. ThusSince $X$ is irreducible, $\Gamma (X,{\mathcal{K}}_X^{\ast })=K^{\ast }$. Under this identification the map to $\Gamma (X,\mathcal{Q})$ is $f\mapsto \mathrm{div}(f)$. Its image is exactly the group of principal divisor elements, and hence

If ${\check{H}}^1(S,\mathcal{O})=0$, that cocycle is a Čech coboundary: there are $f_i\in \mathcal{O}(U_i)$ withSet $F=-f_0$ on $U_0$ and $F=P_m-f_m$ on $U_m$. These formulas agree on overlaps, so the sheaf gluing axiom gives a global meromorphic function. Moreover $F-P_m=-f_m$ is holomorphic near $x_m$. Thus $F$ solves the Mittag-Leffler problem on a Riemann surface with precisely the prescribed principal parts.

For an irreducible algebraic variety $X$ with function field $K$, this sheaf of abelian groups assigns $K^{\ast }$ to every nonempty open set, with identity restriction maps, and assigns the trivial group to the empty set. Irreducibility ensures that any two nonempty open subsets meet, so these sections satisfy the sheaf gluing axiom. It is a flasque sheaf and has zero higher sheaf cohomology.

Every presheaf morphism from $\mathcal{F}$ to a sheaf $\mathcal{H}$ factors uniquely through the canonical sheafification map. Local representatives give local images in $\mathcal{H}$, and the sheaf gluing axiom makes their union unique.