By Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj

Complaints of the second one foreign convention on Cumulative Ring thought held in June 1992 at Fes, Morocco. Paper.

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X By the based homotopy extension property, there is a homotopy ft : X Ñ X with f0 ✏ idX and ft i ✏ i gt . Therefore f1 i ✏ i g1 ✏ ✝, and so f1 induces f1✶ : X ④A Ñ X such that f1✶ q ✏ f1 ✔ f0 ✏ idX , where q : X Ñ X ④A is the projection. Next we show qf1✶ ✔ idX ④A . Since ft i ✏ i gt , the homotopy ft induces a homotopy ht : X ④A Ñ X ④A such that qft ✏ ht q. Then h0 q ✏ qf0 ✏ q, so h0 ✏ idX ④A . Also h1 q ✏ qf1 ✏ qf1✶ q, and so h1 ✏ qf1✶ . Therefore 28 1 Basic Homotopy qf1✶ ✏ h1 ✔ h0 ✏ idX ④A .

We say that the pair ♣X, Aq is a CW pair. 22) and that the CW topology of A coincides with the induced topology of A. The notion of a CW pair can be generalized. 15 A pair of unbased spaces ♣X, Aq with X Hausdorff and A a closed subspace of X is called a relative CW complex if there is a sequence of subspaces of X, called skeleta or relative skeleta, ♣X, Aq0 ❸ ♣X, Aq1 ❸ ☎ ☎ ☎ ❸ ♣X, Aqn ❸ ♣X, Aqn 1 ❸ ☎ ☎ ☎ whose union is X. These subspaces are inductively defined as follows. 1. ♣X, Aq0 is the union of A and a discrete set of points disjoint from A.

Thus ♣Φβ q✁1 ♣F ❳ e¯nβ q is closed in Eβn , for every β. We argue by induction on n that F ❳➨X n is closed in X n . Suppose F ❳ X n✁1 is closed in X n✁1 . If q : X n✁1 ❭ β Eβn Ñ X n is the quotient function, it follows that q ✁1 ♣F ❳ X n q ➨ is closed in X n✁1 ❭ β Eβn . Therefore F ❳ X n is closed in X n . (3) We suppose that C meets an infinite number of open cells of X and choose an infinite set S ✏ tx1 , x2 , . ✉ ❸ C such that each element of S is in a different open cell. For any subset A ❸ S, we show that A is closed in X.