1.7. AUTOMORPHISMS OF FIELDS 17 Theorem 1.6.27 (Heine-Borel). Let S be a closed and bounded subset of R. Given a collection {Ui}i∈I of open sets such that S i∈I Ui, there exists a finite subcollection U1,...,Un of {Ui}i∈I such that S U1 ∪· · ·∪Un. Proof. Suppose that S is a nonempty, closed, bounded subset of R. If a = glb S and b = lub S, then, since S is closed, a and b are in S, and S [a, b]. Let {Ui} be a collection of open sets such that S Ui. By adjoining the complement of S (if necessary), we obtain a collection U of open sets whose union contains [a, b]. Now let B = {x [a, b] | [a, x] is covered by a finite number of open sets in U}. Then B is nonempty since a B, and B is bounded above by b. Let c = lub B. If c = b, we are done. If c b, then there exists y such that c y b and [c, y] is in the same open set that contains c. Thus [a, y] is covered by the same collection of open sets from U that covers [a, c]. This is a contradiction, and hence b must equal c. Thus [a, b] is covered by a finite number of open sets from U, and by throwing away the complement of S (if necessary), S is covered by a finite number of open sets from the original collection. Definition 1.6.28. Let A be a subset of R. An open cover of A is a collection of open sets {Ui}i∈I such that A i∈I Ui. Definition 1.6.29. Let A be a subset of R. We say that A is a compact set if every open covering of A has a finite subcovering. That is, if {Ui}i∈I is an open covering of A, then there is a finite subcollection U1,U2,...,Un of the collection {Ui}i∈I so that A U1 U2 · · · Un. Definition 1.6.30. A subset A of R is sequentially compact if every infinite sequence in A has a subsequence that converges to an element of A. Exercise 1.6.31. Show that a subset of R is compact if and only if it is closed and bounded. The result in the following exercise can be proved easily using the Bolzano- Weierstrass and Heine-Borel theorems in R. We will see in Chapter 2 that the same theorem is true in metric spaces. In the next section, we give an indication of how this works in C. Exercise 1.6.32. A subset of R is compact if and only if it is sequentially compact. 1.7. Automorphisms of Fields For any field F , we can consider the following problem: given f : F −→ F such that f(x + y) = f(x) + f(y) and f(xy) = f(x)f(y) for all x, y F , what can you say about f? Well, if f(x) = 0 for all x F , then it clearly has these properties, but it is not of much use. So, let us assume that there exists an a F such that f(a) = 0.
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