# 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).

$\square$