# Sheaf on basic open sets.

Starting from a $$B$$-sheaf $$\hat{\mathcal{F}}$$, a sheaf defined on the basic open sets of a topological space $$X$$, I am asked to prove that the construction $$\mathcal{F}(U) = \varprojlim_{V \subset U, V \in B} \hat{\mathcal{F}}(V)$$ defines a sheaf.

I already shown that $$\mathcal{F}$$ defines a presheaf, where the restriction maps are given by the universal property of the inverse limit. However, I got stuck trying to show that it satisfies the sheaf axiom.

If a take an open cover $$U = \bigcup_{i \in I}U_i$$ by open sets $$U_i \subset X$$, and I take sections on $$s_i = \mathcal{F}(U_i)$$ such that $$s_i = s_j$$ on $$U_i \cap U_j$$ for all $$i, j$$, I am supposed to show that there is a unique section $$s \in \mathcal{F}(U)$$ such that restricted to $$U_i$$ is exactly $$s_i$$ for all $$i$$.

Now, a section $$s_i \in \mathcal{F}(U_i)$$ is actually a tuple of sections $$(t_V)_V$$ where $$t_V \in \hat{\mathcal{F}}(V)$$ for each $$V \subset U, V \in B$$, and such that the restriction maps satisfy $$\mathrm{res}_{U_i,U_j}(s_j) = s_i$$ for all $$U_i \subset U_j$$.

However, I don't know what to infer from this to prove the sheaf axiom. A hint rather than an answer would be very much appreciated. Thank you.

First, beware, you wrote "the basic open sets of a topological space $$X$$". I do not know if it is just a small mistake while writing, but there can be multiple basis for a given topology. Here we just chose one, that is $$B$$.
Now, to define $$s \in \mathcal{F}(U)$$ is the same as giving compatible sections $$s_V$$ for each $$V$$ a basic open set in $$U$$ (compatible in the following meaning : if $$V_0$$ is a subset of both $$V_1$$ and $$V_2$$, where all of these $$V$$ are basic open sets, then $$s_{V_0}$$ is the restriction of both $$s_{V_1}$$ and $$s_{V_2}$$). Now, given such a $$V \subset U$$, if we want $$s$$ to satisfy the restriction conditions on $$U_i$$, then $$s_V$$ should satisfy certain conditions. Can you see what they are and how to recover $$s_V$$ from them ? I will leave some hints in the following spoiler that you should not read before trying by yourself.
that is, $$s_V$$ restricted to $$V \cap U_i$$ should be equal to $$s_i$$ restricted to $$V \cap U_i$$. Also, notice that you can replace the cover $$V = \bigcup (V \cap U_i)$$ with another cover where any open set of the cover is a basic open set (I will leave this part to you). Now you have restricted your problem to the case where all open sets are basic open, which is exactly the $$B$$-sheaf axiom.
In the end, all you need to show is that the $$s_V$$ you got this way are compatible, so that you can recover $$s \in \mathcal{F}(U)$$. You will also have to show unicity, but the proof is in the same vein :
$$s$$ is uniquely determined by the $$s_V$$, and the $$s_V$$ are unique with the restriction conditions given by the ones on $$s$$.