Let the projective line $\mathbb{P}^1$ be the set of all points $[x:y]$ and suppose $\mathbb{A}^1$ is the subset of all those points with $y \neq 0$. So we may write $\mathbb{A}^1 = \{ [x:1] \mid x \in \mathbb{C} \}$.
I was wondering what the open affine subsets of $\mathbb{A}^1$ are. These are all the affine subsets, so quasi-projective varieties isomorphic to some Zariski closed subset of $\mathbb{A}^n$ somewhere, which are also open. I'm using Smith et al.'s Invitation to Algebraic Geometry. I have the following question: the authors never use Zariski-open affine subset, but are these open affine subsets always Zariski open? I am not sure about it. If this is the case, then I can determine the open affine subsets of $\mathbb{A}^1$ relatively easily.
They have to be Zariski open, so they are either all of $\mathbb{A}^1$, empty or $\mathbb{A}^1$ minus a finite number of points. (Finite complement and Zariski topology coincide on $\mathbb{A}^1$.) The first two are obviously also affine; they are even Zariski closed in $\mathbb{A}^1$. The third class of open subsets are also affine. If $\mathbb{A}^1 \setminus \{a_1, \ldots, a_K \}$ is arbitrary, then the open set $$\mathbb{A}^1 \setminus \{a_1, \ldots, a_K \} = \mathbb{A}^1 \setminus \mathbb{V}( (x-a_1)\cdots(x- a_K) )$$ is an affine algebraic variety by the discussion on p. 53, section 4.2 (on basis for the Zariski topology) in Smith's Invitation.
So I am wondering if this solution is correct? I believe it hinges on the assumption/question I highlighted in the beginning.
To answer my own question, with credits to @ParthShimpi.
The answer is yes. Whenever "open affine" is used, we really mean "Zariski open affine". This is because if we say something is an "open affine subset" of $\mathbb{A}^n$, then the word open here refers to open in the relevant topology on $\mathbb{A}^n$ and the space $\mathbb{A}^n$ is usually defined as the set $\mathbb{C}^n$ equipped with the Zariski topology. This is certainly done very explicitly in Smith's Invitation (see Section 1.2 on page 8 on the Zariski topology), the text book that I am using for learning Algebraic Geometry. Whenever we have "open affine" used together in this way, we can thus be sure this is in reference to the Zariski topology.
So this answers the question fully for me.