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 – IDOK Sep 10 '11 at 16:18
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1http://mathoverflow.net/questions/198/how-do-you-show-that-s-infty-is-contractible – George Sep 10 '11 at 16:23
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George's link was really fabulous than mine ,please look at it, – IDOK Sep 10 '11 at 16:43
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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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This question is similar to: Unit sphere in $\mathbb{R}^\infty$ is contractible?. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – R. H. Vellstra Mar 30 '25 at 01:55