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I'm trying to understand better what Hatcher does in the beginning chapter on cell complexes and so in this sense I would really like someone to elucidate for me what are the subcomplexes of $S^{\infty}$, with the cell structure described in the book, having two cells in each dimension.

The clearer the explanation the better! Thanks

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Have you tried making the list of subcomplexes yourself? For example. What are the 1-dimensional subcomplexes? – Mariano Suárez-Alvarez Feb 8 '13 at 6:32
    
of course i tried; I'm pretty sure the subcomplexes are the $n$-dimensional spheres and the $n$-dimensional spheres, each along with an $n+1$ cell attached to the previous sphere along its boundary. but i was hoping for a detailed answer... – TJIF Feb 8 '13 at 6:47
    
Why don't you write what you did in the question? – Mariano Suárez-Alvarez Feb 8 '13 at 6:50
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@TJIF - do you understand how to make, for example, $S^2$ with two cells in each dimension (less than 3). – Joe Tait Feb 8 '13 at 9:39
    
got it, thanks. – TJIF Feb 21 '13 at 2:45

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