Sections on a compact set extend to a neighbourhood

Reduction to the presheaf value at $K$. The source of the canonical map of the statement is $P(K)$ and the map is the component ${\eta }_{P,K}$, by [F1]; the structure maps of the colimits and the sheafification map are additive (and $k$-linear in the vector-space case), so ${\eta }_{P,K}$ is additive. If $K=\mathrm{\varnothing }$, the empty open set is a terminal stage of the neighbourhood diagram, and both $P(K)$ and $(i^{-1}\mathcal{F})(K)$ are zero by the empty-cover sheaf axiom; hence the assertion holds. For surjectivity below assume $K\mathrm{\ne }\mathrm{\varnothing }$. It therefore suffices to prove that ${\eta }_{P,K}$ is injective and surjective.

Classes and germs. For later use, record the descriptions of [F2]: elements of $P(K)$ are classes $[V,a]$ with $V\supseteq K$ open and $a\in \mathcal{F}(V)$, equality being agreement after restriction to an open $W$ with $K\subseteq W\subseteq V\cap V^{\mathrm{\prime }}$; the stalk $P_x$ at $x\in K$ is the filtered colimit of the $P(U)$ and is identified with ${\mathcal{F}}_x$, and equality of germs in $P_x$ is agreement on a smaller open neighbourhood.

${\eta }_{P,K}$ is injective. Let $p,q\in P(K)$ with ${\eta }_{P,K}(p)={\eta }_{P,K}(q)$; write $p=[V,a]$ and $q=[V^{\mathrm{\prime }},a^{\mathrm{\prime }}]$ with $K\subseteq V,V^{\mathrm{\prime }}$ open and $a\in \mathcal{F}(V)$, $a^{\mathrm{\prime }}\in \mathcal{F}(V^{\mathrm{\prime }})$ by F2. Fix $z\in K$: the germ of ${\eta }_{P,K}(p)$ at $z$ is ${\eta }_{P,z}(p_z)$ and that of ${\eta }_{P,K}(q)$ is ${\eta }_{P,z}(q_z)$ by [F3], and these are equal, so $p_z=q_z$ in $P_z$ because ${\eta }_{P,z}$ is injective; by F2 this means that $a$ and $a^{\mathrm{\prime }}$ agree on an open neighbourhood of $z$. Let $\mathcal{O}$ be the family of all open subsets $O\subseteq V\cap V^{\mathrm{\prime }}$ with $a{\mathrm{\mid }}_O=a^{\mathrm{\prime }}{\mathrm{\mid }}_O$: it is cut out by a formula, and the previous sentence shows that it covers $K$, so by compactness of $K$ ([F5]) there are finitely many $O_1,\dots ,O_m\in \mathcal{O}$ with $K\subseteq O_1\cup \cdots \cup O_m$. Put $W:=O_1\cup \cdots \cup O_m$, an open set with $K\subseteq W\subseteq V\cap V^{\mathrm{\prime }}$; the restrictions $a{\mathrm{\mid }}_W$ and $a^{\mathrm{\prime }}{\mathrm{\mid }}_W$ agree by locality of the sheaf $\mathcal{F}$ ([F4]), since they agree on each $O_t$. Hence $p=[V,a]=[W,a{\mathrm{\mid }}_W]=[W,a^{\mathrm{\prime }}{\mathrm{\mid }}_W]=[V^{\mathrm{\prime }},a^{\mathrm{\prime }}]=q$ by F2, and ${\eta }_{P,K}$ is injective.

Local representatives of a section. Let $\sigma \in (i^{-1}\mathcal{F})(K)$. For $x\in K$ the map ${\eta }_{P,x}:P_x\to (i^{-1}\mathcal{F}{)}_x$ is onto by [F3], so there is $\xi \in P_x$ with ${\eta }_{P,x}(\xi )={\sigma }_x$; by the description of the stalk $P_x$ as a filtered colimit, F2, the element $\xi$ is the germ at $x$ of an element $b\in P(U)$ for some open $U\subseteq K$ containing $x$, and by F2 we can write $b=[V,a]$ with $V\supseteq U$ open in $X$ and $a\in \mathcal{F}(V)$. Then the section ${\zeta }_V(a){\mathrm{\mid }}_U={\eta }_{P,U}(b)$ of $i^{-1}\mathcal{F}$ over $U$ has at $x$ the same germ as $\sigma$ by [F3], and two sections of the sheaf $i^{-1}\mathcal{F}$ with the same germ at $x$ agree on some open neighbourhood $U^{\mathrm{\prime }}\subseteq U$ of $x$ by the description of equality in the stalk (F2 applied to $i^{-1}\mathcal{F}$, [F4]). Form the family $\mathcal{V}$ of all open subsets $U\subseteq K$ for which there are an open $V\subseteq X$ with $U\subseteq V$ and a section $a\in \mathcal{F}(V)$ satisfying ${\zeta }_V(a){\mathrm{\mid }}_U=\sigma {\mathrm{\mid }}_U$: the family is cut out by a formula, and the argument above shows that every point of $K$ lies in a member of $\mathcal{V}$.

Finitely many pieces with compact cores. By compactness of $K$ ([F5]) there are finitely many members $U_0,\dots ,U_m\in \mathcal{V}$ with $K=U_0\cup \cdots \cup U_m$, and for each $i$ we select witnesses: an open $V_i\subseteq X$ with $U_i\subseteq V_i$ and a section $a_i\in \mathcal{F}(V_i)$ with ${\zeta }_{V_i}(a_i){\mathrm{\mid }}_{U_i}=\sigma {\mathrm{\mid }}_{U_i}$; these are finitely many selections ([F5], [F3]). The compact Hausdorff space $K$ is normal ([F5]), so its finite open cover $(U_0,\dots ,U_m)$ can be shrunk: recursively for $i=0,\dots ,m$, maintaining $K=({\bigcup }_{j<i}{B}_j)\cup ({\bigcup }_{j\ge i}{U}_j)$, the set $F_i:=K\setminus ({\bigcup }_{j<i}{B}_j\cup {\bigcup }_{j>i}{U}_j)$ is closed in $K$ and contained in $U_i$, and normality supplies an open $B_i\subseteq K$ with $F_i\subseteq B_i\subseteq \overline{B_i}\subseteq U_i$, the closure taken in $K$ (A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$; the closure is the smallest closed superset and equals $A$ together with its derived set); the invariant is maintained and at the end $K=B_0\cup \cdots \cup B_m$. Put $C_i:=\overline{B_i}$, a compact subset of $K$ with $C_i\subseteq U_i$ ([F5]); then $C_0,\dots ,C_m$ are compact subsets of $K$ with $K=C_0\cup \cdots \cup C_m$ and $C_i\subseteq U_i\subseteq V_i$ for every $i$.

Neighbourhoods of the pairwise intersections. For all indices $i,j$ let ${\mathcal{O}}_{ij}$ be the family of all open subsets $O\subseteq V_i\cap V_j$ with $a_i{\mathrm{\mid }}_O=a_j{\mathrm{\mid }}_O$. It covers $C_i\cap C_j$: for $z\in C_i\cap C_j\subseteq U_i\cap U_j$ the sections ${\zeta }_{V_i}(a_i)=\sigma {\mathrm{\mid }}_{U_i}$ and ${\zeta }_{V_j}(a_j)=\sigma {\mathrm{\mid }}_{U_j}$ have the same germ ${\sigma }_z$ at $z$, so $a_i$ and $a_j$ have the same germ at $z$ by [F3], and by F2 they agree on an open neighbourhood of $z$. Since $C_i\cap C_j$ is a closed subset of the compact set $C_i$, hence compact ([F5]), finitely many members of ${\mathcal{O}}_{ij}$ cover it; their union is an open set $W_{ij}$ with $C_i\cap C_j\subseteq W_{ij}\subseteq V_i\cap V_j$ and, by locality of $\mathcal{F}$ ([F4]), $a_i{\mathrm{\mid }}_{W_{ij}}=a_j{\mathrm{\mid }}_{W_{ij}}$.

Fusion of two pieces. Let $C_1,C_2\subseteq X$ be compact, let $V_1,V_2\subseteq X$ be open with $C_k\subseteq V_k$ and let $s_k\in \mathcal{F}(V_k)$ for $k=1,2$; suppose that $W\subseteq V_1\cap V_2$ is open with $C_1\cap C_2\subseteq W$ and $s_1{\mathrm{\mid }}_W=s_2{\mathrm{\mid }}_W$. Then there are an open $V^{\mathrm{\prime }}\subseteq X$ with $C_1\cup C_2\subseteq V^{\mathrm{\prime }}$ and a section $s^{\mathrm{\prime }}\in \mathcal{F}(V^{\mathrm{\prime }})$, together with open $V_k^{\mathrm{\prime }}\subseteq V_k$ with $C_k\subseteq V_k^{\mathrm{\prime }}$ and $V^{\mathrm{\prime }}=V_1^{\mathrm{\prime }}\cup V_2^{\mathrm{\prime }}$, such that $s^{\mathrm{\prime }}{\mathrm{\mid }}_{V_k^{\mathrm{\prime }}}=s_k{\mathrm{\mid }}_{V_k^{\mathrm{\prime }}}$ for $k=1,2$. Indeed, $A:=C_1\setminus W$ and $B:=C_2\setminus W$ are closed subsets of the compact sets $C_1$ and $C_2$, hence compact, and they are disjoint because $C_1\cap C_2\subseteq W$; as $X$ is Hausdorff there are disjoint open sets $G_1\supseteq A$ and $G_2\supseteq B$ ([F5]). Set $V_1^{\mathrm{\prime }}:=V_1\cap (G_1\cup W)$ and $V_2^{\mathrm{\prime }}:=V_2\cap (G_2\cup W)$: these are open, $C_1=(C_1\cap W)\cup A\subseteq V_1^{\mathrm{\prime }}$ and $C_2=(C_2\cap W)\cup B\subseteq V_2^{\mathrm{\prime }}$, and $V_1^{\mathrm{\prime }}\cap V_2^{\mathrm{\prime }}\subseteq (G_1\cup W)\cap (G_2\cup W)=W\cup (G_1\cap W)\cup (G_2\cap W)\cup (G_1\cap G_2)\subseteq W$, since $G_1\cap G_2=\mathrm{\varnothing }$. Hence $s_1{\mathrm{\mid }}_{V_1^{\mathrm{\prime }}}$ and $s_2{\mathrm{\mid }}_{V_2^{\mathrm{\prime }}}$ agree on $V_1^{\mathrm{\prime }}\cap V_2^{\mathrm{\prime }}$ and glue, by the sheaf axiom for $\mathcal{F}$ ([F4]), to a section $s^{\mathrm{\prime }}$ of $\mathcal{F}$ over $V^{\mathrm{\prime }}=V_1^{\mathrm{\prime }}\cup V_2^{\mathrm{\prime }}\supseteq C_1\cup C_2$ with $s^{\mathrm{\prime }}{\mathrm{\mid }}_{V_k^{\mathrm{\prime }}}=s_k{\mathrm{\mid }}_{V_k^{\mathrm{\prime }}}$ for $k=1,2$.

Iterated fusion. Fuse the pieces of step 1.5 one after another: for each $k\in \{0,\dots ,m\}$ there are an open $V^{(k)}\supseteq C_0\cup \cdots \cup C_k$ and a section $a^{(k)}\in \mathcal{F}(V^{(k)})$ such that ${\zeta }_{V^{(k)}}(a^{(k)})$ and $\sigma$ agree on an open neighbourhood of $D_k:=C_0\cup \cdots \cup C_k$ in $K$. For $k=0$ this is step 1.5. For the inductive step, $D_k$ is compact ([F5]) and on the compact set $D_k\cap C_{k+1}$ the sections ${\zeta }_{V^{(k)}}(a^{(k)})$ and ${\zeta }_{V_{k+1}}(a_{k+1})$ both agree with $\sigma$ on open neighbourhoods, so their germs agree everywhere on $D_k\cap C_{k+1}$ and [F3] gives that $a^{(k)}$ and $a_{k+1}$ have equal germs at every point of it; the compactness-and-locality argument of step 1.6 applied to these two sections produces an open $W\subseteq V^{(k)}\cap V_{k+1}$ with $D_k\cap C_{k+1}\subseteq W$ and $a^{(k)}{\mathrm{\mid }}_W=a_{k+1}{\mathrm{\mid }}_W$. Apply the fusion of step 1.7 with $(C_1,V_1,s_1):=(D_k,V^{(k)},a^{(k)})$ and $(C_2,V_2,s_2):=(C_{k+1},V_{k+1},a_{k+1})$: it gives open $V^{\mathrm{\prime }\mathrm{\prime }}=V_1^{\mathrm{\prime }}\cup V_2^{\mathrm{\prime }}$ and $a^{\mathrm{\prime }\mathrm{\prime }}\in \mathcal{F}(V^{\mathrm{\prime }\mathrm{\prime }})$ with $a^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\mid }}_{V_1^{\mathrm{\prime }}}=a^{(k)}{\mathrm{\mid }}_{V_1^{\mathrm{\prime }}}$, $a^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\mid }}_{V_2^{\mathrm{\prime }}}=a_{k+1}{\mathrm{\mid }}_{V_2^{\mathrm{\prime }}}$, where $D_k\subseteq V_1^{\mathrm{\prime }}$ and $C_{k+1}\subseteq V_2^{\mathrm{\prime }}$. Let $E_k\subseteq V^{(k)}\cap K$ be an open neighbourhood of $D_k$ on which ${\zeta }_{V^{(k)}}(a^{(k)})=\sigma$. By restriction compatibility [F3], ${\zeta }_{V^{\mathrm{\prime }\mathrm{\prime }}}(a^{\mathrm{\prime }\mathrm{\prime }})$ agrees with $\sigma$ on $(V_1^{\mathrm{\prime }}\cap E_k)\cup (V_2^{\mathrm{\prime }}\cap U_{k+1})$, an open neighbourhood of $D_{k+1}$ in $K$; locality [F4] gives agreement on this union. This proves the claim for $k+1$.

${\eta }_{P,K}$ is surjective. In the notation of step 2.1 with $k=m$ we have an open $V:=V^{(m)}\supseteq C_0\cup \cdots \cup C_m=K$ and a section $a\in \mathcal{F}(V)$ such that ${\zeta }_V(a)$ and $\sigma$ agree on an open neighbourhood of $K$; in particular they have equal germs at every point of $K$. Two sections of the sheaf $i^{-1}\mathcal{F}$ over $K$ with equal germs at every point agree by locality ([F4]), so ${\zeta }_V(a)=\sigma$, and since $V\supseteq K$ this says $\sigma ={\eta }_{P,K}([V,a])$ by [F3]. Hence ${\eta }_{P,K}$ is surjective.

The isomorphism and the disjoint-family gluing. By steps 1.1, 1.3 and 3.1 the map ${\eta }_{P,K}$ is bijective and additive, so the canonical map of the statement is an isomorphism of abelian groups (of $k$-vector spaces when $\mathcal{F}$ is a sheaf of $k$-vector spaces). For the moreover clause let $K_1,\dots ,K_m$ be pairwise disjoint compact subsets and $s_j\in \mathcal{F}(U_j)$ with $U_j\supseteq K_j$ open. For $m=0$ take the zero section on the empty open set; for $m=1$ take $s_1$ on $U_1$. For $m\ge 2$, fuse $(K_1,s_1)$ with $(K_2,s_2)$ by step 1.7 taking $W:=\mathrm{\varnothing }$ in the hypothesis, which is an open subset of $V_1\cap V_2$ with $K_1\cap K_2=\mathrm{\varnothing }\subseteq W$ and the vacuous agreement $s_1{\mathrm{\mid }}_{\mathrm{\varnothing }}=s_2{\mathrm{\mid }}_{\mathrm{\varnothing }}$, obtaining open $V_1^{\mathrm{\prime }},V_2^{\mathrm{\prime }}$ with $K_j\subseteq V_j^{\mathrm{\prime }}\subseteq U_j$ and a section $s^{\mathrm{\prime }}\in \mathcal{F}(V_1^{\mathrm{\prime }}\cup V_2^{\mathrm{\prime }})$ equal to $s_j$ on $V_j^{\mathrm{\prime }}$ for $j=1,2$. Now fuse $s^{\mathrm{\prime }}$ with $s_3$ in the same way, with $W:=\mathrm{\varnothing }$ again, and continue; maintaining that the current section equals $s_j$ on an open subset of $U_j$ containing $K_j$ for each $j$ already treated, the fusion step preserves the previous agreements because its new first domain $V_1^{\mathrm{\prime }}$ contains the union of the treated compact sets and the current section equals the previous one on $V_1^{\mathrm{\prime }}$. After $m-1$ fusions this yields an open neighbourhood of $K_1\cup \cdots \cup K_m$ and a section restricting to $s_j$ on an open neighbourhood $W_j\subseteq U_j$ of $K_j$ for every $j$; applying this with the zero section $s_2=0$ over $X$ on a compact set $K_2$ disjoint from $K=K_1$ is the final assertion. The canonical map is additive and is $k$-linear in the vector-space case; the only selections used were finitely many, as recorded in [F5].