- “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).
- For what does there exist a proper subfield such that ?
- 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.