= Infinite-dimensional real projective space
{title2=$\mathbb{RP}^{\infty}$}
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.
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