Struggling with basic facts about schemes. Questions regarding open subsets of schemes. This is data I am thinking about after reading sections 1,2,3 of chapter 2 on schemes from Hartshorne's Algebraic Geometry.
Basically, I know very little and I'm very uncomfortable with schemes.


Let $X$ be a scheme.
We know that every point is in some open affine $U_i \cong \operatorname{Spec}(A_i)$. So, we can cover $X$ be open affines $U_i \cong \operatorname{Spec}(A_i)$.
Now, we can intersect any open subset of $X$ with the cover of open affines.
(1) Does this mean that any open subset of $X$ be can covered (abusing notation) by basic open subsets $D(f_{i_j}) \subset \operatorname{Spec}(A_i)$? Therefore, any point in $X$ is in some (abusing notation) $D(f_{i_j}) \cong \operatorname{Spec}(A_{i_{f_{i_j}}})$?

An exercise shows that any open subset is a scheme via the induced scheme structure.
(2) Does this mean that any cover of $X$ will give us a cover by open affines? For example, take any open subset $U$. Then $U$ is a scheme via the induced scheme structure. So, we can cover $U$ via open affines. Since $U$ is open, then these open affines are also open affines of $X$?

(3) If $p \in X$ is in some open affine $U \cong \operatorname{Spec}(A)$, can we also keep finding smaller and smaller open affines containing $p$? How do these smaller and smaller open affines relate to $U$ and $X$? How do the rings relate to each other?
 A: (1): yes, exactly.
(2): yes. Note that affine-ness for an open subset $U$ of the scheme $X$ doesn’t depend on $X$ itself, only on $U$, apart from $X$ defining the structure sheaf on $U$.
(3): theoretically, yes (if you allow for equalities).
The property is the following: if $U$ is any open subset of a scheme and $p \in U$, there exists an affine open subset $p \in W \subset U$.
But beware, the topology on a scheme isn’t like a Euclidean topology – Zariski open subsets are relatively scarce. In important special cases (local rings, fields) it’s possible that $U$ is a minimal open subset containing $p$, (that is, there is no smaller one). It’s also (that’s rather the opposite phenomenon) possible that any nonempty open subset $U$ contains $p$. This being said, in many examples, you still have enough cases to restrict the open subset further.
Note that it’s rarely useful to ask for a strict restriction – most early algebraic phenomena can be appropriately studied through Zariski open subsets.
