Problems
 “Last night’s temperature took all values between 3 and 5.” Show that it suffices to say “Both 3 and 5 occured among last night’s minimum temperatures (by location).” Assume that temperature is a continuous function of time and location.
 Let be closed under addition. Let be nonincreasing and additive (). Show that on where .
 Let be (additive) abelian groups. For homomorphisms from to , define their sum by . Then the set of homomorphisms from to forms an abelian group. Suppose is a group ( prime).
 Prove that becomes a topological group under the topology by taking to be a neighborhood base around 0.
 Prove that is complete in this topology.
 Prove that every connected component consists of a single element.
 When is compact?
 For what does there exist a proper subfield such that ?
Solutions (Sketches)
 Use intermediate value theorem twice, once for time and once for location.
 Proof follows the same argument as in (Cauchy’s Equation).

 Show that negation is a continuous function and addition is a continuous function . The first is obvious. For the second, note that if then — is an open set in .
 Take the metric defined by where . Given a Cauchy sequence , we define as follows: for with order by the Cauchy condition will eventually stay in the same coset . Then stabilizes; set equal to this value. One can show that is a homomorphism and .
 It suffices to show that each open set is disconnected; it suffices to show that can be written as a union of disjoint closed sets. Just write . (the complements of either can be written as a union of cosets=open sets in the form .)
 If is finite for each , then the proof of compactness is similar to the proof in . Suppose there’s an open cover without a finite subcover; by induction and Pigeonhole Principle take a coset without a finite subcover. Take ; this Cauchy sequence converges to some by completeness. There must be an open set in the cover containing , but it contains for large enough , contradiction.
Else for some . Then the cosets of form an open cover without a finite subcover.
This reduces the problem to a group theory problem…
 Working on it.
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