Sheaves

Which of these presheaves of real-valued functions on $\R$ are sheaves? (Select all that apply.)

How many sections does the sheaf $\underline{\Z/2}$ of locally constant $\Z/2$-valued functions have over $U = (0, 1) \cup (2, 3) \cup (4, 5) \subset \R$?

In the sheaf of continuous functions on $\R$, which pair of functions has the same germ at $0$?

On $X = \C \setminus \{0\}$, the morphism $\exp : \mathcal{O} \to \mathcal{O}^\times$ is surjective on every stalk. Is $\exp : \mathcal{O}(X) \to \mathcal{O}^\times(X)$ surjective?

In the proof that a morphism of sheaves $\varphi : F \to G$ which is bijective on stalks is surjective on sections, local preimages $s^{(x)} \in F(V_x)$ of $t$ are glued. Why do $s^{(x)}$ and $s^{(y)}$ agree on $V_x \cap V_y$?

Cover $S^1$ by three open arcs $U_1, U_2, U_3$ such that each pairwise intersection is a single (connected) arc and the triple intersection is empty. What is $\dim_{\Z/2} \check H^1(\mathcal{U}, \underline{\Z/2})$?

Given a cover U : ι → Opens X of V with hU : ∀ i, U i ≤ V and a compatible family s, which conclusion in the definition of IsSheaf captures both locality and gluing?

Sections of the sheaf of all functions on a finite set are dictionaries. Write glue(sections) that takes a list of dictionaries (functions on subsets) and returns the single dictionary on the union that restricts to each of them, or None if two of them disagree on a common point.