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How can I prove that $S^\infty$ is contractible? Is $S^\infty$ a CW complex?

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see here for some good discussion on proof ,and i think you can surely find it interesting –  Iyengar Sep 10 '11 at 16:18
mathoverflow.net/questions/198/… –  George Sep 10 '11 at 16:23
George's link was really fabulous than mine ,please look at it, –  Iyengar Sep 10 '11 at 16:43
In answer to your second question, $S^\infty$ is a CW complex. Let $X^0=*\coprod *$, and then obtain $X^{n+1}$ from $X^n$ by gluing two copies (the two hemispheres) of $D^{n+1}$ to $X^n$. Then $S^\infty\cong \operatorname{colim} X^n$. –  SL2 Sep 10 '11 at 18:07

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