Stiefel–Whitney class of the universal real line

The base is a CW complex by [F4], hence paracompact Hausdorff CGWH and admissible, and its tautological line is numerable by [F4]. The projective bundle of ${\gamma }_1$ is ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ itself, since a point of $P({\gamma }_1)$ is a line in a line, and its tautological line is ${\gamma }_1$; by [F2] the defining relation is $x_{{\gamma }_1}+w_1({\gamma }_1)=0$ in $H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$.

By [F3], $x_{{\gamma }_1}$ is the pullback of $a$ along any classifying map of ${\gamma }_1$. The identity map of ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ classifies ${\gamma }_1$, so $x_{{\gamma }_1}={\mathrm{id}}^{\ast }a=a$. Substituting in step 1.1 and using that $-1=1$ in ${\mathbb{F}}_2$ gives $w_1({\gamma }_1)=a$; by the rank convention $w_i({\gamma }_1)=0$ for $i\ge 2$ and $w_0=1$, so $w({\gamma }_1)=1+a$.

Boundary cases. The degree-zero class is the unit $1$ in $H^0$ of the connected space ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, matching the degree-zero convention. The bundle is a line, so every Stiefel–Whitney class of degree at least two vanishes, while $w_1=a$ is nonzero by [F1]. All these classes have ${\mathbb{F}}_2$ coefficients. The base is nonempty, so no empty-base convention is exercised, and no choice beyond [A1] is made.