Tautological lines over projective spaces

For finite $N$, by definition $\mathbb{F}{P}^N$ is the quotient of ${\mathbb{F}}^{N+1}\setminus \{0\}$ by nonzero scalar multiplication, so a point is exactly a line $\mathrm{\ell }\subseteq {\mathbb{F}}^{N+1}$. Hence $\mathbb{F}{P}^N={\mathrm{Gr}}_1({\mathbb{F}}^{N+1})$, and the set $\{(\mathrm{\ell },v):v\in \mathrm{\ell }\}$ is exactly the tautological bundle in [F1]. For $N=\mathrm{\infty }$, both projective space and its tautological line are the filtered unions of these finite stages inside ${\mathbb{F}}^{\mathrm{\infty }}$; no expression ${\mathbb{F}}^{\mathrm{\infty }+1}$ is used.

For finite $N$, the displayed bundle inclusion sends its fiber over $\mathrm{\ell }$ to the same line $\mathrm{\ell }$ in ${\mathbb{F}}^{N+1}\subseteq {\mathbb{F}}^{\mathrm{\infty }}$. Taking image planes therefore sends $\mathrm{\ell }$ to itself under the standard finite-stage inclusion, and pulling back ${\gamma }_1$ returns the original pairs $(\mathrm{\ell },v)$. At $N=\mathrm{\infty }$ the same assertion is the identity on the filtered union.

For $N=\mathrm{\infty }$, the finite-stage inclusions unite to the identity on ${\mathrm{Gr}}_1({\mathbb{F}}^{\mathrm{\infty }})$. The classifying-space notation in [F2] gives $B\mathrm{O}(1)$ and $B\mathrm{U}(1)$, while step 1.1 gives $\mathbb{R}{P}^{\mathrm{\infty }}$ and $\mathbb{C}{P}^{\mathrm{\infty }}$. All identifications are direct and use no choice principle.