Topology of Cell Complexes

(3) X = ⋃ n X n with the weak topology: A set A ⊂ X is open (or closed) iff A∩X is open (or closed) in X for each n . Note that condition (3) is superfluous when X is finite-dimensional, so that X = X for some n . For if A is open in X = X , the definition of the quotient topology on X implies that A∩X is open in X , and then by the same reasoning A∩X is open in X , and similarly for all the skeleta X . Each cell e α has its characteristic map Φα , which is by definition the composition D α ֓ X n−1 ∐ αD n α→X n ֓ X . This is continuous since it is a composition of continuous maps, the inclusion X ֓ X being continuous by (3). The restriction of Φα to the interior of D α is a homeomorphism onto e α .