Furstenberg as an Introduction to Topology

2021-10-01

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I've never taken a formal topology course; when I was at university it was considered a graduate-level topic. I've since read some overviews on the subject, but I still only have a cursory understanding.

A few years ago I came across a topological proof of the infinitude of prime numbers, first formulated in 1955 by Hillel Furstenberg^. I find all proofs of the infinitude of the primes interesting, but what caught my eye about this proof is that it also functions as a handy introduction to many basic topological concepts. This note discusses how those topics are introduced in the proof.

We begin by defining a topological space T on the set of integers Z consisting of all arithmetic progressions spanning (-infinity,infinity). Specifically, a subset t∈Z is considered open if and only if it is either empty or a union of arithmetic sequences A(a,b), a≠0, where

A(a,b)={an+b | n∈Z}

Immediately the concepts of a topological space, an open subset, and set union--key concepts in the establishment of topology--are presented to a new reader, exposing him or her to fundamental building blocks of the field.

What is a topological space, anyway? Different equivalent definitions exist, but in the context of Furstenberg's proof, we define a topological space as a set S with a family T of subsets t of S that satisfy the following axioms:

• The empty set Ø and S itself are in T.
• Any union of t_i∈T is also in T.
• Any finite intersection of t_i∈T is also in T.

All t∈T are considered open by definition. The complement of an open set is a closed set. Any union of open sets is open; a finite union of closed sets is closed, but an infinite union not necessarily so. This will be important later.

Does our construction meet the definition of a topology?

• Ø is in T by definition; Z is equivalent to the progression A(1,0), so Z itself is also in T.
• Any union union(t_i) is in T by definition.
• A finite intersection of arithmetic sequences intersect(A(a_1,b_1),A(a_2,b_2),...,A(a_n,b_n)) is either empty (and thus in T) or contains an element m. In the latter case, define a* as the least common multiple of a_1, a_2, ..., a_n; our intersection is then equivalent to A(a*,m).

Verifying these axioms is is a gentle beginner's exercise, giving a new topology student the opportunity to do his or her first critical thinking in the field. In this case the solution requires only elementary arguments--half of the conditions are actually satisfied prescriptively!

There are some interesting properties to note regarding this space.

• Every t∈T is both open and closed. Why? By definition any t is open, so the complement of t is closed. However, given a set t=A(a,b), its complement is

union(A(a,b+1),A(a,b+2),...,A(a,b+(a-1)))

This is a union of sets in T and is thus in T itself, so it must also be open. Therefore the complement of t is open as well as closed, and t itself must be closed as well as open.

• Every t∈T is either empty or an infinite arithmetic sequence, and by definition, these are the only open sets in Z. Therefore a finite set of integers cannot be an open set, and its complement cannot be a closed set.

Along with learning definitions, the proof allows us to explore the properties of open and closed sets, which gives some exposure to the nature of sets in topology. A student thus gains direct experience in contemplating topological constructs.

We now consider the primes as a set in T. Suppose the primes are finite and denote the largest as p. Consider

P=union(A(q,0),q prime)=union(A(2,0),A(3,0),...,A(p,0))

P is a finite union of closed sets, so P itself is closed. However, P includes all multiple of all prime numbers; this yields all integers except -1 and 1, so the set {-1,1} is the complement of P. As shown earlier, this finite set cannot be open; thus P cannot be closed. This is a contradiction. □

The majority of the setup for this proof involves establishing very basic topological concepts. In fact, once those concepts are understood, the final section of the proof is almost a direct consequence of considering the set of prime arithmetic sequences. Meanwhile, everything from a topological space to an open set to the behavior of unions of sets is examined as part of the proof. The first few sections of any topology course can be easily summed up by examining it; it could even serve as a good extra-credit question on an initial quiz or exam.

^ Furstenberg, Harry. "On the Infinitude of Primes." The American Mathematical Monthly 62, no. 5 (1955): 353. doi:10.2307/2307043.

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