How can I prove that $S^\infty$ is contractible? Is $S^\infty$ a CW complex?
Is $S^\infty$ contractible? Does it have CW structure?
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algebraic-topology
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0see [here](http://www.math.niu.edu/~rusin/known-math/93_back/s-infty) for some good discussion on proof ,and i think you can surely find it interesting – 2011-09-10
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1http://mathoverflow.net/questions/198/how-do-you-show-that-s-infty-is-contractible – 2011-09-10
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0George's link was really fabulous than mine ,please look at it, – 2011-09-10
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0In 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 $S^\infty\cong \operatorname{colim} X^n$. – 2011-09-10