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Recall $$S^\infty = \cup S^n,\ {\bf RP}^\infty = \cup {\bf RP}^n,\ {\bf CP}^\infty = \cup {\bf CP}^n$$

Hence $S^\infty / {\bf Z}_2 ={\bf RP}^\infty$. And I think that the following is possible : ${\bf CP}^\infty = \cup S^{2n+1}/S^1=S^\infty /S^1$.

Here how can we obtain the natural quotient map ${\bf RP}^\infty \rightarrow {\bf CP}^\infty $ ?

Thank you in advance.

HK Lee
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    Just take the quotient map $S^\infty \rightarrow \mathbb{C}P^\infty$ and note that it identifies antipodal points, hence descends to a map $\mathbb{R}P^\infty \rightarrow \mathbb{C}P^\infty$. – Dylan Wilson Jun 11 '13 at 15:11

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