I'm just starting to learn algebraic geometry for the first time, and I had an elementary question concerning some of my intuitions about open subschemes (during this period where I haven't yet learnt the methods needed to prove them).
Let $X = \mathbb{A}_k^n = \text{Spec}(k[\textbf{x}])$ be affine space of dimension $n$ over $k$, where $k$ is an algebraically closed field, and let $\mathcal{O}_X$ be the structure sheaf on $X$.
Claim: If $U \subseteq X$ be an open set which is not contained in any non-trivial basic open set (i.e. an open set of the form $U_f = X \setminus \mathbb{V}(f)$ for some $f \in k[\textbf{x}]$ s.t. $U_f \neq X$), then $\mathcal{O}_U(U) = \mathcal{O}_X(X) = k[\textbf{x}]$.
Question: Is the claim and the following reasoning correct?
My reasoning is that $\mathcal{O}_U(U) = \mathcal{O}_X(U)$ seems to correspond to the ring obtained by localising $k[\textbf{x}]$ at every element which is always non-zero on the open set $U$. If $U$ is not contained in a non-trivial basic open set, then by definition, there is no non-unit which is always non-zero* on $U$. Hence, $\mathcal{O}_U(U)$ is only trivially localised, so $\mathcal{O}_U(U) = k[\textbf{x}]$.
*Noting that every constant polynomial is a unit because $k$ is a field, and every non-constant polynomial in the variables $\textbf{x}$ has a root in $k$ since $k$ is algebraically closed. If $k$ was not an algebraically closed field, I believe that these intuitions would break down.
You are correct and in fact you remarked an instance of an interesting general phenomenon.
Let $A$ be a noetherian normal(=integrally closed in its fraction field) domain. Let $S$ be the set of primes $\mathfrak{p}\in Spec(A)$ of height $1$. The following seemingly boring fact of commutative algebra holds: $$ A=\bigcap_{\mathfrak{p}\in S} A_{\mathfrak{p}} $$ were this intersaction makes sense as localizations of domains can be thought of as subrings of the fraction field.
You should look at the nontrivial inclusion from right to left as saying the following, surprising geometric fact: elements of Frac$A$ (" meromorphic functions on the affine scheme $Spec A$") that are defined over all codimension one agebraic subsets are actually globally defined. In other words, there exists only poles in codimension 1: a meromorphic function defined on the complement of a codimension $\geq 2$ subset can always be extended to the whole set. Vakil calls this "algebraic Hartogs lemma" since something very similar hold under the name of Hartogs' Lemma in the analytic setting.
In your situation, you indeed have a Noetherian normal scheme, so you can apply this lemma over any affine neighbourhood. Any open set that is not contained in one of the form $D(f)$ must be the complement of a closed set of codimension $\geq 2$, hence sections over this open can be extended local piece by local piece to sections of $\mathcal{O}_X (X)$ :)
See Vakil, 11.3.11 in the latest version of the notes.