By Samuel Eilenberg, Norman Steenrod

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I g [CHAP. 5 2 x,. = d 2 Xi Then u = (w 2 d, 2 ). It follows now that MAYER-VIETORIS SEQUENCE 15] = hi+u fore = \l/u (6) = hi^dzXi /fci^Wj hi^Ui = vi. 41 = Similarly h 2 ^u v2 There- . v. u If e H (A) and $(u) q exactness, there exists x a H e g = 0, then h^u = ^(X ai A) (a = 1,2) and h^u = 0. By such that This implies By exactness, there is w a e H Q+l (X) such that Then since ki+ V = Ji^i 2j|c THEOREM Y ** ** 1 . 4. Y V Vv-^J -" 2? 3. 2 into the analogous diagram for (F;Fj,F 2 ), it is seen that the desired commutativity relations follow from standard relations.

Although triangulable spaces appear to form a rather narrow class, a major portion of the spaces occurring in applications of topology to geometry and analysis are of this type. Furthermore, it is shown in Chapter x that any compact space can be expressed as a limit of triangulable spaces in a reasonable sense. In this sense, triangulable spaces are dense in the family of compact spaces. of all 2. 1. vertices, usually (1) = I>G4) A a(A)^Q.

1) (2) (3) Explicitly: IfH (X,A) = Ofweaehq,ihmi+: //Hf (X,B) =0 for each q,thm d: IfH Q (A,B) = for each q, then j+: 9 any one of p Conversely , H (X ,B) for each J9T,(A,B) H (X,A) - H^(A B) far each H (X,B) ~ H (X,A) for each q. q 9 9 q. q. q q the three conclusions implies the corresponding hypothesis. 2 except for notation. Let (X,A,B) be an admissible triple. 5. (X) for q all q, then (A,B) B C A C (X,B) Simiq. C (X,B) off g. f PROOF. For the first assertion, the hypothesis and the exactness of = for all q.