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By Tammo tom Dieck

This e-book is a jewel– it explains vital, priceless and deep subject matters in Algebraic Topology that you simply won`t locate in other places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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The Prufer topology makes A into a topological group. This topology is discrete exactly if A itself satisfies the minimum condition. 1) A contains a subgroup of type p". The embedding of Z ( p " ) in the group of complex numbers z with / z /= 1 induces a nondiscrete Hausdorff topology on Z(p"), and by translations one obtains a nondiscrete topology on A . 0 EXERCISES 1. A [ n ] is closed in any topological group A . 2. (a) Prove that every homomorphism between groups is continuous in the p-adic, Priifer, and finite index topologies.

Examples for functors are abundant. The most important ones in abelian groups are those which assign to a group a subgroup o r a quotient group [they are discussed in the next section], and the functors Hom, Ext, 0 , and Tor [defined in Chapters VIII-XI. The following example is of a different type. L ( [ A ] ,[ B ]E 7 ) [ B ] : B1 A+Bz Lz+ B, commute. Pis a category. d as follows. 7,let H [ A ]= Ker x2/Im m I and let H [ y l ,y z , y 3 ] : H [ A ]+ H [ B ] be the homomorphism PI ( aeK er I t is evident that: ( 1 ) a E Ker mz implies y z a t Ker pL;(2) a.

Let F , be subfunctors of the identity where CJ ranges over all ordinals less than an ordinal z. Define the infinite product .. F , . . F I F O(a < z), and show that this is likewise a subfunctor of the identity. * By making use of the notion of direct sums [see 81 verify the formula for a subfunctor F of the identity. Show that the analog fails to hold for direct products. 7. TOPOLOGIES IN GROUPS In abelian groups, topology can be introduced in various ways which are natural in one sense or another.

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