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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 forthe classifying space for $C_2$-equivariant complex line bundles. The$RO(C_2)$-graded Bredon ordinary cohomology was calculated by Gaunce Lewis, buthere we extend to a larger grading in order to capture a more natural set ofgenerators. These generators include the Euler class of the tautologicalbundle, which lies outside of the $RO(C_2)$-graded theory.

The $RO(\Pi B)$-graded $C_2$-equivariant ordinary cohomology of $B_{C_2} U(1)$