TITLE: homotopy pushout of spaces homotopic to finite CW complexes
QUESTION [7 upvotes]: Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW complex?

REPLY [4 votes]: I don't have a reference, but here is an easier argument, based, like John's, on the homotopy
invariance of the homotopy pushout.
The invariance implies that you can replace the maps $i: A\to B$ and $j: A\to C$ with homotopic maps and get the same homotopy pushout, up to homotopy type.  So assume they are cellular.  It is easy to give the inclusions $A\hookrightarrow M_i$ and $A\hookrightarrow M_j$ finite CW structures so that $A$ is a subcomplex of each, and then the union $M_i \cup_A M_j$ inherits a finite CW structure.