I think this is done inductively on the skeletons but I can't work out the details.
How do you prove a CW complex is locally path connected
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general-topology
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3They're not just locally path-connected, they're locally contractible. There's a key theorem about CW-complexes, that the inclusion of any of any subcomplex into the entire CW-complex is a cofibration. Look at that proof and the neighbourhoods constructed in that proof. That should give you the idea for how to prove what you want to prove. – 2011-06-19
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0Please give full details in the question, not just the title. Also, what is a CW complex? – 2011-06-19
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5@Asaf What additional details do you want, exactly? And all definitions are easily googlable. – 2011-06-19
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1@Grigory: It's not that I complain about lack of definitions. I complain about bad formatting, while at it I was asking what is a CW complex. – 2011-06-19
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1@Asaf: First google hit: [CW complex](http://en.wikipedia.org/wiki/CW_complex): A space obtained by gluing disks together. The topologist's preferred notion of a polyhedron. All reasonable (geometric) spaces are CW complexes. C: closure finite, W: weak topology. Inventor: J.H.C. Whitehead. – 2011-06-19
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0Have you looked at Hatcher's book? – 2011-06-19
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0Dear @RyanBudney I saw this problem in Lee's book where he haven't introduce the conception of contractible. Can you tell me how to prove it directly? Thank you! – 2014-07-17
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0I proved this very carefully here https://math.stackexchange.com/questions/1448789/understanding-construction-of-open-nbds-in-cw-complexes/1462419#1462419 – 2017-06-18