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I think this is done inductively on the skeletons but I can't work out the details.

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They'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. –  Ryan Budney Jun 19 '11 at 5:56
    
Please give full details in the question, not just the title. Also, what is a CW complex? –  Asaf Karagila Jun 19 '11 at 6:00
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@Asaf What additional details do you want, exactly? And all definitions are easily googlable. –  Grigory M Jun 19 '11 at 6:07
    
@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. –  Asaf Karagila Jun 19 '11 at 6:13
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@Asaf: First google hit: 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. –  t.b. Jun 19 '11 at 11:12
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