You can "visualize" the cell structure on $\mathbb{R}P^n$ rather explicitly as follows. The set of tuples $(x_0, ... x_{n+1}) \in \mathbb{R}^{n+1}$, not all equal to zero, under the equivalence relation where we identify two tuples that differ by multiplication by a nonzero real number, can be broken up into pieces depending on which of the $x_i$ are equal to zero.

• If $x_0 \neq 0$, the corresponding points can be written $(1, x_1, ... x_{n+1})$, and they form a subspace isomorphic to $\mathbb{R}^n$.

• If $x_0 = 0$ and $x_1 \neq 0$, the corresponding points can be written $(0, 1, x_2, ... x_{n+1})$, and they form a subspace isomorphic to $\mathbb{R}^{n-1}$.

And so forth. One way to say this is that the tuples where $x_0 \neq 0$ form an affine slice or affine cover of $\mathbb{R}P^n$ and the tuples where $x_0 = 0$ constitute the "points at infinity," which themselves form a copy of $\mathbb{R}P^{n-1}$.

You can "visualize" the cell structure on $\mathbb{R}P^n$ rather explicitly as follows. The set of tuples $(x_0, ... x_n) \in \mathbb{R}^{n+1}$, not all equal to zero, under the equivalence relation where we identify two tuples that differ by multiplication by a nonzero real number, can be broken up into pieces depending on which of the $x_i$ are equal to zero.

• If $x_0 \neq 0$, the corresponding points can be written $(1, x_1, ... x_n)$, and they form a subspace isomorphic to $\mathbb{R}^n$.

• If $x_0 = 0$ and $x_1 \neq 0$, the corresponding points can be written $(0, 1, x_2, ... x_n)$, and they form a subspace isomorphic to $\mathbb{R}^{n-1}$.

And so forth. One way to say this is that the tuples where $x_0 \neq 0$ form an affine slice or affine cover of $\mathbb{R}P^n$ and the tuples where $x_0 = 0$ constitute the "points at infinity," which themselves form a copy of $\mathbb{R}P^{n-1}$.