  We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients,
of $CP_{C_2}^\infty = B_{C_2} U(1)$, the complex projective space, a model for
the classifying space for $C_2$-equivariant complex line bundles. The
$RO(C_2)$-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but
here we extend to a larger grading in order to capture a more natural set of
generators. These generators include the Euler class of the tautological
bundle, which lies outside of the $RO(C_2)$-graded theory.
