# A natural section of the Etale space

Let $$P: (OP_{X}) \to (Sets)$$ be a sheaf, where $$(OP_X)$$ is just the category of open sets of a topological space $$X$$. Define $$P_x$$ to be the stalk of $$x \in X$$. Then we can define the Etale space to be $$Et(P):=\amalg_{x \in X} P_x$$, and the topology is given by a basis $$\{U_s| U \ open\ in X\}$$, where $$U_s = \{S_x|x \in U\}$$($$S_x$$ is the germ of $$S$$ at $$x$$). There is a natural projection $$\pi: Et(p) \to X$$, which is continuous and surjective.

Now the claim is that there is a natural continuous map $$g: X \to Et(p)$$ such that $$\pi \circ g = id_{X}$$.

My approach is to associate with each point $$x \in X$$ a section $$s \in P(X)$$, since $$X$$ is a natural open set containing $$x$$. However this choice of section seems to be too arbitrary to be continuous.

The reason that I ask about this is that the existence of a global section ensures that we can define $$P^{+}$$ the sheaf associated to $$P$$ because $$P^{+}(U)$$ is defined to be the set of all continuous sections on $$U$$ and the restriction of a global section onto each open subset ensures that $$F^{+}(U)$$ is not empty.

This answer gives a more precise account of this construction

Update: Inspired by the given link, given $$s \in P(X)$$ we can consider directly the map $$g:X \to Et(P)$$, $$g(x) = s_x$$. Now I want to show that it is continuous. $$g(x) \in U_{s'}$$ iff there is an open set $$U'$$ containing $$x$$ such that $$r^{X}_{U'}s = r^{U}_{U'}s'$$. What then?

• "natural"? Associated to each global section of the sheaf there such a map $g$. Indeed such maps $g$ correspond exactly to the global sections of the sheaf (so when there are no global sections, there are no such $g$). Commented Jul 4, 2019 at 2:52
• I added something in the question. If a global section does not exist then $F^{+}$ cannot always be defined which seems wrong to me. @LordSharktheUnknown Commented Jul 4, 2019 at 3:16
• The definition of the sheaf $P^+(U)$ uses sections on every open subset $U$, not necessarily global sections. If for any $U$ there is no sections on $U$, well, then you have the empty presheaf and its sheafification will be the empty sheaf. But if $P$ has no global section, it might have sections on smaller $U$ and the sheafification won't necessarily be empty. Commented Jul 4, 2019 at 8:08