By Michael Atiyah

Those notes are according to the process lectures I gave at Harvard within the fall of 1964. They represent a self-contained account of vector bundles and K-theory assuming purely the rudiments of point-set topology and linear algebra. one of many good points of the therapy is that no need is made from traditional homology or cohomology idea. actually, rational cohomology is outlined by way of K-theory.

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Those notes are in response to the process lectures I gave at Harvard within the fall of 1964. They represent a self-contained account of vector bundles and K-theory assuming in basic terms the rudiments of point-set topology and linear algebra. one of many gains of the remedy is that no need is made from usual homology or cohomology thought.

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4. Assume that the '1ft are inner • premeasures 'v't E I. Then 'If is inner • tight. Thus 'lfx for nonvoid K C I is inner • tight as well. Proof. Fix A C B in 'I, and then U E j(l) such that A,B E 'Ih. l. some D c Q \ P and M C Q 'v'M E VJl such that 'l'u(Q)- 'l'u(P)- E < inf{'l'u(M) : ME VJ'l}. l. u) :ME VJ'l}. This proves the assertion. Next we return to section 2 in order to obtain the conclusion that 'I' is • continuous at 0 . 34 have the obvious consequence which follows. 5. Assume that the 'I1 are • compact and hence the ljf1 are • continuous at Vt E I.

The resultant one-to-one correpondence α ⇔ β was then extended to arbitrary Hausdorff spaces X in Schwartz [1973], and used to define Borel-Radon measures on 43 ¨ KONIG 110 these spaces. Thus the definition of these measures came under the restriction of local finiteness, which meanwhile thanks to Berg-Christensen-Ressel [1984] could be abandoned. However, it is clear that in the present context conditions of the type of local finiteness are natural ones. In the present classical example it is nontrivial to note that α = β can happen even for locally compact X.

6. Assume that the ljf1 are • continuous at 0 Vt E I. j. m C 'I~ for some nonvoid countable K C I. Dl ljf(M) = 0. Proof. 2). I) Let I be countable, so that K =I= N. m} is> 0 and hence >some E > 0. 1). Thecasen = 1: We haveljf= 'l't x '1'{2,. {2,-··})*. m} >E. The step 1 ;£ n:::} n + 1: We have 'i'{n+l,-··} = 'l'n+l X 'i'{n+2,-··} on 'I{n+l,-··} = ('In+l X 'I{n+2,-··})*. m} > E, where M(at, ···,an)(an+ I)= M(at, .. · ,an,an+t) is obvious from the definition. This terminates the inductive choice.