Infinite real projective space is a marked mod-two Eilenberg–Mac Lane space

In the published models, $S^{\mathrm{\infty }}=V_1({\mathbb{R}}^{\mathrm{\infty }})$ and ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}={\mathrm{Gr}}_1({\mathbb{R}}^{\mathrm{\infty }})$. Send a unit vector to its line. Over the open set of lines with nonzero coordinate $j$, choose the unique unit representative having positive $j$th coordinate; the other representative is its negative. These maps give two disjoint local sections and an evenly covered open set. Their formulas are continuous on every finite stage and therefore for the weak Grassmannian/Stiefel topologies. These open sets cover the base, so this is a two-sheeted covering with antipodal deck involution and discrete fiber $\{+1,-1\}$.

The published finite-local-strip covering-homotopy lemma makes this a Hurewicz fibration. Its total space is contractible by stable Stiefel contractibility; it is in particular connected and simply connected. The fibration long exact sequence gives ${\pi }_j({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }})=0$ for $j\ge 2$, since positive homotopy groups of the fiber vanish. Path lifting associates to a based loop its endpoint deck transformation. This is a group homomorphism: lifting successive loops composes their endpoint deck transformations. It is surjective since a path from a unit vector to its negative exists in the connected sphere, and injective since a loop whose lift closes is nullhomotopic in the contractible total space and its contraction projects to the base. Thus ${\pi }_1=\mathbb{Z}\mathrm{/}2={\mathbb{F}}_2$, with the stated marking. The published Schubert theorem gives the CW base. This proves the marked Eilenberg–Mac Lane claim.

The published projective-space cohomology lemma gives a unique nonzero degree-one class $x$ and the ring ${\mathbb{F}}_2[x]$. The normalized fundamental class is nonzero by the published Eilenberg–Mac Lane representability theorem; uniqueness in degree one therefore identifies it with $x$. Marked Eilenberg–Mac Lane uniqueness transfers this ring to any chosen marked $K_1$.