 Infinite-dimensional real projective space

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is $\mathbb{Z}$ in degree zero, $\mathbb{Z}/2$ in positive even degrees and zero in odd degrees. Its mod-two cohomology is ${\mathbb{Z}}_2$ in every nonnegative degree.