Attaching Cells to Homotopy Equivalent Spaces

We shall use the following standard fact of algebraic topology, which we state here with only a sketch of a proof. 

Under these conditions, the space Xi Uf a is homotopy equivalent to the space X2 U/2 7. 

The following two special cases of Theorem 11.11 are often distinguished as being of particular importance  

2 The Main Theorem of Discrete Morse Theory for CW Complexes 

Intuitively, a cellular collapse is a strong deformation retract that pushes the interior of a maximal cell in, using one of its free boundary cells as the starting point, much like compressing a body made of clay. The cellular collapses can be defined for arbitrary CW complexes. 

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