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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?

  • 2
    This is proposition 4 in [the appendix on CW complexes](http://www.math.cornell.edu/~hatcher/AT/ATapp.pdf) to Hatcher's Algebraic Topology.2011-10-05

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