Editing Real projective space (section)

==Infinite real projective space==
The infinite real projective space is constructed as the [[direct limit]] or union of the finite projective spaces:
<math display="block">\mathbf{RP}^\infty := \lim_n \mathbf{RP}^n.</math>
This space is [[classifying space for O(n)|classifying space of ''O''(1)]], the first [[orthogonal group]].

The double cover of this space is the infinite sphere <math>S^\infty</math>, which is contractible. The infinite projective space is therefore the [[Eilenberg–MacLane space]] ''K''('''Z'''<sub>2</sub>, 1).  

For each nonnegative integer ''q'', the modulo 2 homology group <math>H_q(\mathbf{RP}^\infty; \mathbf{Z}/2) = \mathbf{Z}/2</math>.  

Its [[cohomology ring]] [[modulo (jargon)|modulo]] 2 is 
<math display="block">H^*(\mathbf{RP}^\infty; \mathbf{Z}/2\mathbf{Z}) = \mathbf{Z}/2\mathbf{Z}[w_1],</math>
where <math>w_1</math> is the first [[Stiefel–Whitney class]]: it is the free <math>\mathbf{Z}/2\mathbf{Z}</math>-algebra on <math>w_1</math>, which has degree 1.

Its [[cohomology ring]] with <math>\mathbf{Z}</math> coefficients is
<math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) = \mathbf{Z}[\alpha]/(2\alpha), </math>
where <math>\alpha</math> has degree 2. It can be deduced from the [[chain map]] between cellular cochain complexes with <math>\mathbf{Z}</math> and <math>\mathbf{Z}/2</math> coefficients, which yield a ring homomorphism 
<math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) \rightarrow H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math>
injective in positive dimensions, with image the even dimensional part of <math>H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math>. Alternatively, the result can also be obtained using the [[Universal coefficient theorem]].