 Assume inductively that "f" is cellular on the ("n" − 1)-skeleton of "X", and let "e""n"  be an "n"-cell of "X".The closure of "e""n" is compact in "X", being the image of the characteristic map of the cell, and hence the image of the closure of "e""n" under "f" is also compact in "Y".Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is, intersects non-trivially) only finitely many cells of the complex.