From: ikastan@alumnae.caltech.edu (Ilias Kastanas)
Newsgroups: sci.math
Subject: Re: strange topology on the positive integers
Date: 8 Mar 1996 10:48:57 GMT
In article <4hi4q8$fo9@itsmail1.hamilton.edu>,
Rick Decker wrote:
>esouche@ens-lyon.fr (Estelle Souche) wrote:
>>
>>A friend of mine recently found a strange topology on
>>the positive integers:
>>let E(a,b)={n | n positive integer, b divides (n-a)}
>>The sets E(a,b) where a and b are relavitely prime,
>>are a basis of this topology.
>>
>>With this topology, N* (the set of positive integers)
>>is Hausdorff and connected, the sets E(a,b) are
>>totally discontinuous (I'm not sure it's the right
>>English term.)
>>
>>My friend conjectures that with this topology, the
>>set of all prime numbers is connected. But he hasn't
>>found a proof until now.
>>
>>Have you ever read anything about this topology?
>>Or do you have some ideas about this conjecture?
>>
>This topology has been part of mathematical folklore for some time.
>It was the subjects of some posts in this group last year and at that
>time no one came up with an original citation. It can be
>used to provide an amusing proof of the infinitude of primes:
>
>For each prime, p, let S_p = Union(i = 1, p-1) E(i, p). Observe
>that {1} cannot be open, since open sets in this topology are
>all infinite. Note that {1} = Intersection(all primes p) S_p, so
>if the set of all primes was finite, we'd have an immediate
>contradiction.
>
>Here's what (apparently) was known about this topology last year.
>
>1. The space is Hausdorff, as you noted, but not regular.
>2. Any finite set is closed.
>3. P, the set of all primes, is neither open nor closed. The set
> has empty interior and is nowhere dense in N*. It is not
> compact.
>4. The space is neither compact nor connected. In fact, the space
> is totally disconnected, which answers your friend's conjecture
> in the negative. The space is second countable by definition,
> hence it's Lindelof and separable.
There are two variants of this topology, a finer one where (a, b)
are not required to be relatively prime, and a coarser one where b must
be prime. Confusion is easy!
It is true the space is not compact... not even locally compact.
But it _is_ connected; it is a nice example of a countable connected
Hausdorff space. It fails to be locally connected, though. (The coarser
topology mentioned is both connected and locally connected).
It is true that the space is not regular; in fact any two open sets
have intersecting closures. By the way, the regular open sets (i.e. those
that are the interiors of their closures) are a basis.
P has empty interior (so it is not open!). It is also not closed,
and therefore not compact. It is not connected anyway (as another posting
pointed out, primes not of the form 1 + 12k are either of the form 2 + 3m
or 3 + 4n). But I don't think P is nowhere dense; far from it!
Ilias