Understanding homogeneous coordinates

Question

I've seen that one can decompose $\mathbb{R}P^2$ into a 0-cell, 1-cell and 2-cell:

$\mathbb{R}P^2 = \{[1:0:0] \} \cup \{[x:1:0] : x \in \mathbb{R}\} \cup \{[x:y:1]: x,y \in \mathbb{R}\}$

I have a couple questions about this:

1. Is $ \{[x:y:0] : x,y \in \mathbb{R}\} = \{[x:y:1] : x,y \in \mathbb{R}\}$ since as I understand the former is just the collection of points $(x,y)$, whereas the latter is all the lines $y = \lambda x$. If the difference is to do with points at infinity, how does $\{[x:1:0] : x \in \mathbb{R}\}$ capture these? Or are they the same and the latter is preferred just to make the cell 2 dimensional?
2. Is the correct decomposition of $\mathbb{R}P^n$ into $1, \dots, n$ cells:

$$\begin{align} \mathbb{R}P^n = &\{[1:0:\dots:0] \}\\ &\cup \{[x_1:1:0:\dots:0] : x_1 \in \mathbb{R}\} \\& \cup \{[x_1:x_2:0:\dots:0:1] : x_1,x_2 \in \mathbb{R}\} \\& \cup \vdots \\&\cup \{[x_1:x_2:\dots:x_n:1] : x_1,x_2, \dots x_n \in \mathbb{R}\} \end{align}$$

Answer 1

1. Wrong. Every homogeneous coordinate represents a line through the origin. Such coordinates can equivalently be seen as representing points. These points are where the lines intersect a fixed plane (not situated at the origin), which we'll call the "screen". If the line is parallel to the screen then the point of intersection is understood to be "at infinity". The points at infinity form a projective subspace one dimension lower than the finite points.
2. Yes, except your first factor represents a $0$-cell, not a $1$-cell. Thanks to Arctic Char for noticing this.

TL;DR: The convention is that the homogeneous coordinate $[x:y:1]$ is understood to represent the Cartesian coordinate $(x,y)$, while the homogeneous coordinate $[x:y:0]$ is understood to represent a point that's at infinity.