Complex projective plane: $\mathbb C^2$ vs $\mathbb C^3$

Question

I have a question about Phillip A. Griffiths - Introduction to algebraic curves.

In Chapter I.1, it seems to say $\mathbb C^3 \cup L_{\infty} = P^2\mathbb C$, but later on it says $P^2\mathbb C \setminus L_{\infty} = \mathbb C^2$.

What's going on please?

Guesses:

1. I'm wrong: It's not that 'together' means '$\cup$'.

2. I'm wrong: It's not that $\setminus$ means 'set minus'. (Quotient?)

3. I'm right, and so is the book: 'Together' means '$\cup$' and $\setminus$ means 'set minus'. However, no one ever said $\mathbb C^3 \cap L_{\infty} = \emptyset$, so when we take off $L_{\infty}$ from $P^2\mathbb C$, there's still something to take out of $\mathbb C^3$...sooo actually, $\mathbb C^3 \setminus L_{\infty} = \mathbb C^2$ because $L_{\infty} = $ (or $\cong$) $\mathbb C$ or something.

4. I'm right, but the book: is wrong: it should be both $\mathbb C^2$ or both $\mathbb C^3$.

Answer 1

The answer is (4). There is a typographical error in the first statment it should read $$\mathbb{C}^2 \cup L_{\infty} = P^2\mathbb{C}$$