CW - complexes Stephen

The term CW-complex comes from “closure-finite with the weak topology”, where “closure-finite” refers to A3 and “weak topology” refers to A4. A finite complex is a CW-complex with only finitely many cells. Observe that if X is a finite complex, A4 is redundant, since W is the union of the compact sets φα((φ n α) −1W ), and these are closed since X is Hausdorff. If X has cells of dimension n but no cells of higher dimension, we say that X is n-dimensional. X is infinite-dimensional if there are cells of arbitrarily large dimension. A subcomplex A of X is a closed subspace which is a union of cells of X. It is clear that A is then itself a CW-complex, whose characteristic maps are just the given characteristic maps for those cells of X which lie in