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currently taking my first class in homotopy theory and have just seen the definition of cw complex for the first time. im still a bit confused about how the construction works when there are infinitely many cells of the same dimension. suppose $n$ is fixed and $B_\epsilon(a) \subset \mathbb{R}^n$ is the ball about $a$ of the radius $\epsilon$. can anyone explain how the space $\cup_k B_{2^{-k}}((2^{-k},0,\cdots,0))$ is formally constructed as a cw complex?

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    Let $a_k = (2^{-k}, 0, \ldots, 0)$, then $|a_k - a_{k+1}| = 2^{-(k+1)}$. So if $|y-a_{k+1}|\le 2^{-(k+1)}$ then, $|y-a_k| \le |y-a_{k+1}| + |a_k - a_{k+1}| \le 2^{-(k+1)} + 2^{-(k+1)} = 2^{-k}$. In other words, your space is just $B_{2^{-k}}(a_k)$ for the smallest value of $k$ in the union, which is trivially a CW complex. Maybe you meant the sphere? In that case you have a generalized Hawaiian earring which is a classic example of a non-CW-complex. You can see the Hawaiian earring is not a CW complex by proving it is not locally contractible.2012-04-19

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