In R.O.Wells book "Differential Analysis on Complex Manifolds" p. 44 proof of Theorem 2.2 part b) the author claims that any two sections of an etale space which agree at a point agree in some neighborhood of that point (etale space is a (possibly non-Hausdorff) $Y$ with the surjection $p\colon Y \rightarrow X$ which is a local homeomorphism). Why is that true?



Sections of an etale space
First note that if $p\_1:Y\_1 \to X$ and $p\_2:Y\_2 \to X$ are two etale spaces over $X$ then
any morphism of etale spaces $f:Y\_1 \to Y\_2$ (where by morphism of etale spaces I mean that $f$ is continuous and that
$p\_2\circ f = p\_1$) makes $Y\_1$ an etale space over $Y\_2$. 


(Proof: Let $y \in Y\_1$, and let $V$ be a n.h. of $f(y)$ in $Y\_2$ such that $p\_2:V \to
p\_2(V)$ is a homeo. Note that a local homeo. is open by definition, and so $p\_2(V)$
is also open in $X$. Now $f^{-1}(V)$ is an open subset of $Y\_1$ containing $y$,
and so we may find a n.h. $U$ of $y$ in $Y\_1$, contained in $f^{-1}(V)$, such that $p\_1: U \to p\_1(U)$ is a homeo.
Since $p\_1 = p\_2\circ f$, and since $f(U) \subset V$, we find that $f: U \to f(U)$ is
a homeo. and that $f(U)$ is open in $Y\_2$. Thus $f$ is a local homeo., as required.)


Now suppose that $f\_1,f\_2:Y\_1 \rightarrow Y\_2$ are two morphisms of etale spaces,
and that $f\_1(y) = f\_2(y)$ for some $y \in Y\_1$. Then $f\_1$ and $f\_2$ coincide in some
n.h. of $y$.


(Proof: Choose some sufficiently small n.h. $V$ of $f\_1(y) = f\_2(y)$ in $Y\_2$
such that $p\_2:V \to p\_2(V)$ is a homeo. Then choose a n.h. $U$ of $y$ contained 
in $f\_1^{-1}(V) \cap f\_2^{-1}(V)$ such that $f\_i:U \to f\_i(U)$ is a homeo. for
both values of $i$. Since each $f\_i(U)$ is contained in $V$, we see
that $p\_2: f\_i(U) \to p\_2\circ f\_i(U) = p\_1(U)$ is a homeo. as well, and hence
that $p\_1 = p\_2\circ f\_i: U \to p\_1(U)$ is a homeo.


We then find that, on $U$, the map $f\_i$ (for either value of $i$) can be computed
as the composite of $p\_1:U \to p\_1(U)$ and $p\_2^{-1}: p\_1(U) \to f\_i(U) \subset V$.
Thus the two maps $f\_i$ coincide on $U$.)


Now if $p:Y \to X$ is an etale space and $\sigma$ is a section, we can think of $\sigma$
as a morphism from the etale space $X \to X$ (i.e. $X$ thought of as the trivial etale
space over itself) to the etale space $Y \to X$. The preceding result then shows that
if two sections coincide at a point, they coincide in a n.h. of that point.


