Idea of Proof
The proof can be given by induction after n, with the statement that f is cellular on the skeleton Xn. For the base case n=0, notice that every path-component of Y must contain a 0-cell. The image under f of a 0-cell of X can thus be connected to a 0-cell of Y by a path, but this gives a homotopy from f to a map, which is cellular on the 0-skeleton of X.
Assume inductively that f is cellular on the (n − 1)-skeleton of X, and let en be an n-cell of X. The closure of en is compact in X, being the image of the characteristic map of the cell, and hence the image of the closure of en 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. Thus f(en) meets at most finitely many cells of Y, so we can take to be a cell of highest dimension meeting f(en). If, the map f is already cellular on en, since in this case only cells of the n-skeleton of Y meets f(en), so we may assume that k > n. It is then a technical, non-trivial result (see Hatcher) that the restriction of f to can be homotoped relative to Xn-1 to a map missing a point p ∈ ek. Since Yk − {p} deformation retracts onto the subspace Yk-ek, we can further homotope the restriction of f to to a map, say, g, with the property that g(en) misses the cell ek of Y, still relative to Xn-1. Since f(en) met only finitely many cells of Y to begin with, we can repeat this process finitely many times to make miss all cells of Y of dimension larger than n.
We repeat this process for every n-cell of X, fixing cells of the subcomplex A on which f is already cellular, and we thus obtain a homotopy (relative to the (n − 1)-skeleton of X and the n-cells of A) of the restriction of f to Xn to a map cellular on all cells of X of dimension at most n. Using then the homotopy extension property to extend this to a homotopy on all of X, and patching these homotopies together, will finish the proof. For details, consult Hatcher.
Read more about this topic: Cellular Approximation
Famous quotes containing the words idea of, idea and/or proof:
“My fortune somewhat resembled that of a person who should entertain an idea of committing suicide, and, altogether beyond his hopes, meet with the good hap to be murdered.”
—Nathaniel Hawthorne (18041864)
“I had the idea that there were two worlds. There was a real world as I called it, a world of wars and boxing clubs and childrens homes on back streets, and this real world was a world where orphans burned orphans.... I liked the other world in which almost everyone lived. The imaginary world.”
—Norman Mailer (b. 1923)
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other twoa proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)