Hatcher 0.23

Hatcher, Algebraic Topology, Chapter 0

23. Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible.

Proof. Let X be the union of CW complexes A and B which are contractible and for which A\cap B is contractible. Contract the intersection to get the wedge of the complexes A/(A\cup B) and B/(A\cup B). Both of these are contractible (details).

   \square

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