The mod-two Euler class is the top Stiefel–Whitney class

The line case. Let $L\to B$ be a numerable real line bundle over a paracompact Hausdorff CGWH base of CW homotopy type and let $c:B\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ be a classifying map supplied by [F8], so $c^{\ast }{\gamma }_1\cong L$. For the universal line, $P({\gamma }_1)\cong {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ and its tautological line identifies with ${\gamma }_1$ itself. The identity therefore classifies this line, so [F8] gives $w_1({\gamma }_1)={\mathrm{id}}^{\ast }a=a$. The unit sphere of ${\gamma }_1$ is $S^{\mathrm{\infty }}$ by $v\mapsto (\mathbb{R}v,v)$, with inverse the vector projection; these maps are continuous in each finite-stage chart and compatible with the weak colimits. Compute its Euler class: by [F5] the Gysin sequence of the double cover $S({\gamma }_1)\cong S^{\mathrm{\infty }}\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ contains the exact piece $H^0({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)\stackrel{⌣e_2({\gamma }_1)}{\to }{H}^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)\stackrel{{\pi }^{\ast }}{\to }{H}^1(S^{\mathrm{\infty }};{\mathbb{F}}_2);$ the last group vanishes by [F6], so $⌣e_2({\gamma }_1)$ is surjective onto the nonzero one-dimensional group $H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)={\mathbb{F}}_2\cdot a$; hence its value on $1$ is $e_2({\gamma }_1)=a=w_1({\gamma }_1)$. Now naturality of the Euler class [F2] and of $w_1$ [F3] along $c$ gives $e_2(L)=e_2(c^{\ast }{\gamma }_1)=c^{\ast }{e}_2({\gamma }_1)=c^{\ast }a=w_1(L)\mathrm{.}$

The general case by splitting. Let $n\ge 1$ and let $q:\mathrm{Fl}(E)\to B$ be the flag bundle of [F4], with $q^{\ast }E\cong L_1\oplus \cdots \oplus L_n$ and $q^{\ast }$ injective. Naturality [F2] gives $e_2(q^{\ast }E)=q^{\ast }{e}_2(E)$, the product formula for Euler classes [F2] applied to the successive summands gives $e_2(q^{\ast }E)={\prod }_{j=1}^n{e}_2(L_j)$, and step 1.1 turns each factor into $w_1(L_j)$. The Whitney formula [F3] gives $w_n(q^{\ast }E)=w_n(L_1\oplus \cdots \oplus L_n)={\prod }_{j=1}^n{w}_1(L_j)$. Hence $q^{\ast }{e}_2(E)=q^{\ast }{w}_n(E)$, and injectivity of $q^{\ast }$ yields $e_2(E)=w_n(E)$. For $n=0$ both classes are the unit by [F1] and [F3].

The integral clause. Suppose $E$ carries an integral orientation $o$. Reduce an integral cocycle representing its normalized Thom class $u_E$ valuewise modulo two. On relative cochains postcomposition with $\mathbb{Z}\to {\mathbb{F}}_2$ commutes with the differential: $\rho (\phi \bar{\mathrm{\partial }})=(\rho \phi )\bar{\mathrm{\partial }}$. It preserves cocycles and coboundaries, hence by [F7] this defines the coefficient-reduction class ${\rho }_2(u_E)$ and commutes with restriction to every fiber. On a fiber pair $(D^n,S^{n-1})$, the integral normalization is a generator of $H^n(D^n,S^{n-1};\mathbb{Z})\cong \mathbb{Z}$, whose reduction is the unique nonzero element of $H^n(D^n,S^{n-1};{\mathbb{F}}_2)\cong {\mathbb{F}}_2$. Indeed the relative cellular complex for $(D^n,S^{n-1})$ has one generator in degree $n$ and no other generators (also for $n=0$, with $S^{-1}=\mathrm{\varnothing }$); coefficient reduction is $\mathbb{Z}\to {\mathbb{F}}_2$ on that generator by [F9], sending either integral generator to $1$. Thus ${\rho }_2(u_E)$ is normalized for the reduced orientation and equals the normalized mod-two Thom class by uniqueness [F7]. Coefficient reduction also commutes with the pair map $j^{\ast }$ and zero-section pullback, again by the cochain formula in [F7]. Applying the defining composites and step 2.1 gives ${\rho }_2(e(E,o))={\rho }_2(s^{\ast }{j}^{\ast }{u}_E)=s^{\ast }{j}^{\ast }{\rho }_2(u_E)=e_2(E)=w_n(E)$. This uses no orientation hypothesis beyond the existence of the integral orientation; when $E$ is not integrally orientable the second clause is not asserted.

Boundary cases. In rank zero the canonical mod-two orientation is $1$, so $e_2=w_0=1$. A supplied integral cohomological orientation is a locally constant sign, and its Euler class is that sign because $j$ and $s$ are identities; reduction sends either sign to $1$ as in step 3.1. Thus the second assertion also reads $1=1$ after reduction. In rank one the flag projection is an identity up to its canonical bundle isomorphism and step 1.1 applies; every line here has a classifying map by [F8]. For the empty base both sides are the zero class, with the zero ring's unit coinciding with zero. The canonical mod-two orientation is preserved by every bundle isomorphism; integral orientation choices do not enter the first clause. AC is inherited from the classification, Thom, Gysin, splitting and characteristic-class interfaces.