 Bockstein on infinite real projective space

For the coefficient sequence $0\to {\mathbb{Z}}_2\to {\mathbb{Z}}_4\to {\mathbb{Z}}_2\to 0$, the Bockstein homomorphism on infinite-dimensional real projective space is zero from an even degree and an isomorphism from an odd degree. In cellular cohomology, lift the mod-two generator to $1\in {\mathbb{Z}}_4$. The coboundary is zero in even degree and two in odd degree; identifying two with the image of $1\in {\mathbb{Z}}_2$ gives the formula.