Ag.algebraic Geometry – Error in Hartshorne Exercise II.1.1?

It's really a basic question, but let me spell it out. Exercise 1.1 of the second chapter of Hartshorne's algebraic geometry asks to prove that the sheaf associated with the pre-leaf sending each open subset of a topological space $$X$$ to a fixed abelian group $$A$$ is the sheaf of continuous functions of the open subsets of $$X$$ at $$A$$, or $$A$$ is equipped with discrete topology.

I usually prove this fact in the following way. First writing $$mathcal {A}$$ for sending presheaf $$U subseteq X$$ open non-empty subset to $$A$$, and $$mathcal {A ^ prime}$$ the sending of the wreath $$U$$ to the set of continuous functions $$f: U rightarrow A$$ or $$A$$ has the discrete topology. Now I define a map $$theta: mathcal {A} (U) rightarrow mathcal {A ^ prime} (U)$$ who sends the item $$a in A$$ to the constant function $$mathbf {a}: U rightarrow A$$ send each point of $$U$$ at $$a$$. Then, if I have a map of presheaves $$phi: mathcal {A} rightarrow mathcal {G}$$, I discuss this way. Let me consider $$f in mathcal {A} ^ premium (U)$$, and write $$U = bigsqcup_ {i in I} U_i$$, where the $$U_i$$are the connected components of $$U$$. Then define $$a_i: = f (U_i)$$, which is well defined by continuity, then defines $$psi_ {| U_i}: mathcal {A} ^ { prime} (U_i) rightarrow mathcal {G} (U_i)$$ By sending $$f_ {| U_ {i}}$$ at $$b_i: = phi (a_i)$$. By the property of sheaf of $$mathcal {G}$$, all these sections stick together to define a section $$b in mathcal {G} (U)$$. Now, just define $$psi_ {| U} (f) = b$$.

Now, what's wrong with that? Well, the problem is that at one point I define morphism $$psi_ {U}$$ via his restriction to $$U_i$$. The problem is that without any hypothesis on the topological space that I consider, maybe $$mathcal {A} ^ prime (U_i)$$ is not even defined. In fact, if $$X$$ is arbitrary, nothing says that the connected components are open. Think about the situation of $$mathbb {Q}$$ with the usual topology. Therefore, if I assume a condition of $$X$$, for example. that it's connected locally, my evidence works, but what if I suppose that $$X$$ is arbitrary (as does Hartshorne)?