Let $X:=\{p,q_1, q_2\}$ be a topological space with topology $\mathcal{T}:=\{\emptyset, \{p\}, \{q_1,p\}, \{q _2, p\}, X\}$. Define a sheaf $\mathcal{O}$ by $$\mathcal{O}(\emptyset) := \{0\}, \quad \mathcal{O}(\{p\}) := k(x), \quad \mathcal{O}(\{p, q_1\})= \mathcal{O}(\{p, q_2\}) = \mathcal{O}(X) := (k[x])_{(x)}$$ with the evident restritions morphisms. Then is it true that?
$$\mathcal{O}_p \cong k(x), \quad \mathcal{O}_{q_1} \cong \mathcal{O}_{q_2} \cong (k[x])_{(x)} $$
Attempt: Yes, it is true, for example every element of the stalk $\mathcal{O}_p$ has a unique representant of the form $(\{p\}, f)$ with $f \in k(X)$ and we see that the multiplication of such classes is exactly the same multiplication of the field $k(x)$. More formally,
$$ k(x) \to \mathcal{O}_p: f \mapsto [(f,\{p\})] $$ is an isomorphism of rings.
Is this correct?