Sheaves of sets are equivalent to local homeomorphisms over the base space

For a sheaf $\mathcal{F}$, let $p:E(\mathcal{F})\to X$ be as in [F1]. If $\xi =s_x\in E(\mathcal{F})$, then $\xi \in [s,U]$ for some section $s\in \mathcal{F}(U)$. By construction $p:[s,U]\to U$ is bijective. Its inverse $x^{\mathrm{\prime }}\mapsto s_{x^{\mathrm{\prime }}}$ is continuous because for any smaller basic open $[t,V]\subseteq [s,U]$, the preimage is the open set $\{x^{\mathrm{\prime }}\in U\cap V:s_{x^{\mathrm{\prime }}}=t_{x^{\mathrm{\prime }}}\}$, which is open by the sheaf locality encoded in [L1]. Hence $p{\mathrm{\mid }}_{[s,U]}$ is a homeomorphism onto the open set $U$. Since every point of $E(\mathcal{F})$ lies in some $[s,U]$, $p$ is a local homeomorphism.

Let $q:E\to X$ be a local homeomorphism, and let ${\mathrm{Sec}}_q(U)$ denote its continuous sections over an open set $U$. Restriction of a section is again a section, so ${\mathrm{Sec}}_q$ is a presheaf. If two sections of ${\mathrm{Sec}}_q(U)$ agree on an open cover, they are equal pointwise, so locality holds. If ${\sigma }_i\in {\mathrm{Sec}}_q(U_i)$ are compatible on an open cover $U={\bigcup }_i{U}_i$, define $\sigma (x)={\sigma }_i(x)$ for $x\in U_i$. Compatibility makes this well defined, and continuity is local on the cover because each ${\sigma }_i$ is continuous. Thus [L1] holds, so ${\mathrm{Sec}}_q$ is a sheaf.

A sheaf morphism $\phi :\mathcal{F}\to \mathcal{G}$ induces a map over $X$ $E(\phi ):E(\mathcal{F})⟶E(\mathcal{G}),s_x⟼({\phi }_U(s){)}_x\mathrm{.}$ This is well defined by restriction compatibility. It is continuous: if $E(\phi )(s_x)$ lies in a basic open $[t,V]$, equality of the two germs lets us shrink to an open $W\subseteq U\cap V$ on which ${\phi }_U(s){\mathrm{\mid }}_W=t{\mathrm{\mid }}_W$; then $[s{\mathrm{\mid }}_W,W]$ is a neighbourhood of $s_x$ mapped into $[t,V]$. Identities and compositions are preserved. Conversely, a map $f:E\to E^{\mathrm{\prime }}$ over $X$ between local homeomorphisms sends a section $\sigma$ to $f\circ \sigma$, naturally in the open set, and hence induces a sheaf morphism $\mathrm{Sec}(f)$.

For a sheaf $\mathcal{F}$ and an open set $U$, send $s\in \mathcal{F}(U)$ to the section ${\theta }_U(s):U⟶E(\mathcal{F}),{\theta }_U(s)(x)=s_x\mathrm{.}$ By step 1.1 this is continuous. If ${\theta }_U(s)={\theta }_U(t)$, then $s_x=t_x$ for all $x\in U$, so [L1] gives $s=t$. Conversely, let $\sigma :U\to E(\mathcal{F})$ be a continuous section. For each $x\in U$, choose a basic open $[s_x,V_x]$ containing $\sigma (x)$. Continuity makes $W_x:=U\cap V_x\cap {\sigma }^{-1}([s_x,V_x])$ an open neighbourhood of $x$. Since $p$ is injective on $[s_x,V_x]$ and $p\circ \sigma ={\mathrm{id}}_U$, the restriction $\sigma {\mathrm{\mid }}_{W_x}$ equals $y\mapsto (s_x{)}_y$. These local sections agree on overlaps because they induce the same map $\sigma$. By [L1], they glue to a unique $s\in \mathcal{F}(U)$ with ${\theta }_U(s)=\sigma$. Thus ${\theta }_U$ is a bijection, natural in $U$.

For a local homeomorphism $q:E\to X$, define $\eta :E⟶E({\mathrm{Sec}}_q)$ by sending $e\in E$ to the germ at $x=q(e)$ of any local section through $e$. Such a local section exists because $q$ is a local homeomorphism, and two choices have the same germ because on a smaller common chart they are both the inverse of $q$. On a chart $W\stackrel{\sim }{\to }q(W)$, the map $\eta$ is a homeomorphism from $W$ onto the basic open defined by the inverse section. Hence $\eta$ is an isomorphism over $X$. The formula $E(\phi )(s_x)=({\phi }_U(s){)}_x$ makes the bijections $\theta$ of step 2.1 natural in $\mathcal{F}$, while the formula $\eta (f(e))=(f\circ \sigma {)}_x$ makes $\eta$ natural in $E$. Thus the two functors constructed in steps 1.2 and 1.3 are quasi-inverse equivalences.