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?
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2$\begingroup$ Why is this tagged "soft-question"? $\endgroup$– Charles StaatsCommented May 5, 2010 at 3:51
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$\begingroup$ I thought it should be trivial, since the author assumes it so casually... $\endgroup$– nonameCommented May 5, 2010 at 8:39
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$\begingroup$ The soft-question tag is used more on this site for "metamathematical" questions (not sure if that's the right word). I've removed it. $\endgroup$– j.c.Commented May 5, 2010 at 17:18
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1 Answer
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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.
