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**Example text**

Thus (X, {X k }) is a CW complex. The converse is clear from the previous propositions. 15. 14. Then (X, {X n }) is a ﬁnite CW complex. In particular, X is compact. 16. Let x ∈ S n−1 . Let e1 = {x} and e2 = S n−1 − {x}. This gives S n−1 the structure of a CW complex with one 0-cell e1 and one (n − 1)-cell ◦ e2 . Now consider B n ⊃ S n−1 . Let e3 = B n − S n−1 . This makes B n into a CW complex with the 0-cell e1 , the (n − 1)-cell e2 and an n-cell e3 . Note that all this makes sense when n = 1.

Write q : en → C(¯ g (en )) for the restriction of g¯. If m ≤ n, the next step can be omitted, so suppose m > n. Since • • • • q(e n ) ⊂ Y n−1 , q −1 (d − d) ⊂ en − e n . 2, there is a homotopy ◦ • g (en )) − d, and after ﬁnitely many such of q, rel e n , to a map of en into C(¯ •n homotopies, rel e , we will obtain a map of en into Y n . ◦ ◦ If q(en ) ∩ d = ∅, there is nothing to do. Otherwise, let y ∈ q(en ) ∩ d. ◦ ◦ q −1 (y) is a compact set lying in e n . Let D be a neighborhood of y in d which is homeomorphic to a closed ball, and let U = q −1 (int D).

When n > 0 the boundary operator n ∂ : Sn (X; R) → Sn−1 (X; R) is deﬁned by ∂(σ) = (−1)i (σ ◦ Fi ); when i=0 n ≤ 0, ∂ = 0. The graded R-module {Sn (X; R)} is abbreviated to S∗ (X; R). One shows that ∂ ◦ ∂ = 0, so (S∗ (X; R), ∂) is a chain complex over R called the singular chain complex of X. Its singular homology modules are denoted 3 by H∗∆ (X; R). The ring R is the ring of coeﬃcients. A map f : X → Y induces a chain map S∗ (f ) := {f# : S∗ (X; R) → S∗ (Y ; R)} deﬁned by f# (σ) = f ◦ σ. We have4 (g ◦ f )# = g# ◦ f# , and id# = id.