Although topology made away with metric properties of shapes, it was helped very much by algebra in classification of knots. Following is a wonderful example (due to Harry Fürstenberg of the Hebrew University of Jerusalem, Israel) of a returned favor (albeit on a smaller scale): Euclid's theorem - one of the basic statements of arithmetic - is proven by very simple topological means!

We call a set closed if it contains all its near points. A set is open if its complement is closed. There are always two sets (the empty set and the space itself) that are simultaneously open and closed. Open and closed sets are also characterized by complementary properties:

Let's prove the right two. The left ones will follow from de Morgan's laws,

(A)c = Ac, and (A)c = Ac,

Statement 1

Any intersection of closed sets is closed.

Proof

Let point p be near F_t, where F_t's are closed sets for all values of parameter t from some set T. This means that every neighborhood of p has a nonempty intersection with F_t, and, therefore, with every F_t. Since the latter are closed sets, pF_t for all t. Thus pF_t.

Statement 2

A finite union of closed sets is closed.

Proof

Let p be near F_t, where tT a finite set. p must be near at least one of F_t's. For, if it were not the case, for every t there would exist a neighborhood of p that did not intersect F_t. From the way the neighborhoods were defined, there would exist a neighborhood (take the smallest of the aforementioned neighborhoods) of p that did not intersect any of F_t's, nor would it intersect the union F_t in contradiction with nearness of p to F_t. Hence p is near one of F_t's. Therefore, p belongs to that F_t, and, finally, pF_t.

There are various ways to define a topology on a given space. One may start with neighborhoods and deduce from their properties properties of open and closed sets or those of the operation of closure. It's also possible to start with Statements 1 and 2 and the stipulation that the empty set and the whole space are closed, and define neighborhoods and open sets with all the expected properties of the latter. Of course, one can start with open sets as well. And, finally, neighborhoods must not be defined as balls in a metric space. The statements below capture the most important properties of the neighborhoods:

Any family of sets that satisfy N1-N4 can be used to define closed and open sets and other attributes of a topology. We now look at Fürstenberg's example.

Let Z be the set of all integers - positive, negative, and 0. For a, bZ, b > 0 let

Na, b = { a + nb: n Z }.

Each N_{a, b} is a two-sided arithmetic progression. Note also that aN_{a, b} for every b. "N" in the definition is supposed to remind us of neighborhoods. Indeed, think of N_{a, b} as those basic neighborhoods that are neighborhoods of each of their points (N4). Add to the collection of neighborhoods all supersets of N_{a, b}'s. With this definition we only need to verify property N3. But note that N_{a, b}N_{a, c} = N_{a, lcm(b, c)}, from which N3 follows easily.

Call a set U open if, for every aU, there exists b > 0, such that N_{a, b}U. We can check that Statements 1 and 2 hold as are their analogues for the open sets. Two facts are important:

The first of these follows from the definition, the second from N_{a, b} = Z /N_{a+i, b}, where the union is taken over i = 1, 2, ..., b-1. N_{a, b} is then closed as a complement of a finite union of open sets.

Now the punch line. Except for ±1 and 0, all integers have prime factors. Therefore each is contained in one or more N_{0, p}, where p is prime. We thus arrive at the identity:

Z /{-1, 1} = N0, p,

where union is taken over the set {p} = P of all primes. If the latter were finite, the right hand side would be closed as the union of a finite number of closed sets (O2). The set {-1, 1} would then be open as a complement of a closed set. This would contradict O1.

Reference

1. M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 2000
2. H. Fürstenberg, On the Infinitude of Primes, Amer. Math. Monthly 62 (1955), 353
3. K. Janich, Topology, UTM, Springer-Verlag, 1984

Copyright © 1996-2008 Alexander Bogomolny