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I was trying to understand the notion of CW complex from wikipedia. The very first non-example is:

$$\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$$

This is not a CW complex, supposedly because it is not locally contractible.

Could someone help me with why a CW complex(as defined in the wikipedia page) has to be locally contractible?

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This is proposition 4 in the appendix on CW complexes to Hatcher's Algebraic Topology. – Chris Eagle Oct 5 '11 at 18:47

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