Arithmetic progression topologies explained
In general topology and number theory, branches of mathematics, one can define various topologies on the set
of
integers or the set
of positive integers by taking as a
base a suitable collection of
arithmetic progressions, sequences of the form
or
\{...,b-2a,b-a,b,b+a,b+2a,...\}.
The open sets will then be unions of arithmetic progressions in the collection. Three examples are the
Furstenberg topology on
, and the
Golomb topology and the
Kirch topology on
. Precise definitions are given below.
Hillel Furstenberg introduced the first topology in order to provide a "topological" proof of the infinitude of the set of primes. The second topology was studied by Solomon Golomb[1] and provides an example of a countably infinite Hausdorff space that is connected. The third topology, introduced by A.M. Kirch,[2] is an example of a countably infinite Hausdorff space that is both connected and locally connected. These topologies also have interesting separation and homogeneity properties.
The notion of an arithmetic progression topology can be generalized to arbitrary Dedekind domains.
Construction
Two-sided arithmetic progressions in
are subsets of the form
where
and
The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:
(aZ+b)\cap(cZ+b)=\operatorname{lcm}(a,c)Z+b,
where
is the
least common multiple of
and
[3] Similarly, one-sided arithmetic progressions in
are subsets of the form
aN+b:=\{an+b:n\inN\}=\{b,a+b,2a+b,...\},
with
and
. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:
(aN+b)\cap(cN+d)=\operatorname{lcm}(a,c)N+q,
with
equal to the smallest element in the intersection.
This shows that every nonempty intersection of a finite number of arithmetic progressions is again an arithmetic progression. One can then define a topology on
or
by choosing a collection
of arithmetic progressions, declaring all elements of
to be open sets, and taking the topology generated by those. If any nonempty intersection of two elements of
is again an element of
, the collection
will be a
base for the topology. In general, it will be a
subbase for the topology, and the set of all arithmetic progressions that are nonempty finite intersections of elements of
will be a base for the topology. Three special cases follow.
The Furstenberg topology, or evenly spaced integer topology,[4] on the set
of integers is obtained by taking as a base the collection of all
with
and
The Golomb topology,[1] or relatively prime integer topology,[5] on the set
of positive integers is obtained by taking as a base the collection of all
with
and
and
relatively prime.
[1] Equivalently,
[6] the subcollection of such sets with the extra condition
also forms a base for the topology.
[5] The corresponding
topological space is called the
Golomb space.
[7] The Kirch topology,[2] or prime integer topology,[8] on the set
of positive integers is obtained by taking as a
subbase the collection of all
with
and
prime not dividing
[9] Equivalently,
[6] one can take as a subbase the collection of all
with
prime and
.
[2] [8] A
base for the topology consists of all
with relatively prime
and
squarefree (or the same with the additional condition
). The corresponding topological space is called the
Kirch space.
[9] The three topologies are related in the sense that every open set in the Kirch topology is open in the Golomb topology, and every open set in the Golomb topology is open in the Furstenberg topology (restricted to the subspace
). On the set
, the Kirch topology is
coarser than the Golomb topology, which is itself coarser that the Furstenberg topology.
Properties
The Golomb topology and the Kirch topology are Hausdorff, but not regular.[5] [8]
The Furstenberg topology is Hausdorff and regular.[4] It is metrizable, but not completely metrizable.[4] [10] Indeed, it is homeomorphic to the rational numbers
with the
subspace topology inherited from the
real line.
[11] Broughan
[11] has shown that the Furstenberg topology is closely related to the
-adic completion of the rational numbers.
Regarding connectedness properties, the Furstenberg topology is totally disconnected.[4] The Golomb topology is connected,[5] [1] [12] but not locally connected.[5] [12] The Kirch topology is both connected and locally connected.[8] [2] [12]
The integers with the Furstenberg topology form a homogeneous space, because it is a topological ring — in some sense, the only topology on
for which it is a ring. By contrast, the Golomb space and the Kirch space are topologically rigid — the only self-
homeomorphism is the trivial one.
[7] [9] Relation to the infinitude of primes
See main article: Hillel Fürstenberg's proof of the infinitude of primes.
Both the Furstenberg and Golomb topologies furnish a proof that there are infinitely many prime numbers.[1] A sketch of the proof runs as follows:
- Fix a prime and note that the (positive, in the Golomb space case) integers are a union of finitely many residue classes modulo . Each residue class is an arithmetic progression, and thus clopen.
- Consider the multiples of each prime. These multiples are a residue class (so closed), and the union of these sets is all (Golomb: positive) integers except the units .
- If there are finitely many primes, that union is a closed set, and so its complement is open.
- But every nonempty open set is infinite, so is not open.
Generalizations
The Furstenberg topology is a special case of the profinite topology on a group. In detail, it is the topology induced by the inclusion
, where
is the
profinite integer ring with its profinite topology.
The notion of an arithmetic progression makes sense in arbitrary
-
modules, but the construction of a topology on them relies on closure under intersection. Instead, the correct generalization builds a topology out of
ideals of a
Dedekind domain.
[13] This procedure produces a large number of countably infinite, Hausdorff, connected sets, but whether different Dedekind domains can produce homeomorphic topological spaces is a topic of current research.
[14] [15] References
- .
- Book: Steen . Lynn Arthur . Lynn Arthur Steen . Seebach . J. Arthur Jr. . J. Arthur Seebach, Jr. . Counterexamples in Topology . Counterexamples in Topology . 1978 . . Berlin, New York . Dover reprint of 1978 . 978-0-486-68735-3 . 507446 . 1995 .
Notes and References
- Golomb . Solomon W. . 1959 . A Connected Topology for the Integers . 2309340 . The American Mathematical Monthly . 66 . 8 . 663–665 . 10.2307/2309340 . 0002-9890.
- Kirch . A. M. . February 1969 . A Countable, Connected, Locally Connected Hausdorff Space . The American Mathematical Monthly . 76 . 2 . 169–171 . 10.1080/00029890.1969.12000163 . 0002-9890.
- Steen & Seebach, p. 82, counterexample #60, item 1
- Steen & Seebach, pp. 80-81, counterexample #58
- Steen & Seebach, pp. 82-84, counterexample #60
- Web site: The Kirch topology is the same as the prime integer topology .
- Banakh . Taras . Spirito . Dario . Turek . Sławomir . 2021-10-28 . The Golomb space is topologically rigid . 1912.01994 . Commentationes Mathematicae Universitatis Carolinae . 62 . 3 . 347–360 . 10.14712/1213-7243.2021.023 . 240183836 . 0010-2628.
- Steen & Seebach, pp. 82-84, counterexample #61
- Banakh . Taras . Stelmakh . Yaryna . Turek . Sławomir . 2021-12-01 . The Kirch space is topologically rigid . 2006.12357 . Topology and Its Applications . 304 . 107782 . 10.1016/j.topol.2021.107782. 219966624 .
- Lovas . R. . Mező . I. . Some observations on the Furstenberg topological space . Elemente der Mathematik . 70 . 103–116 . 2015 . 3 . 10.4171/EM/283 . 126337479 .
- Broughan . Kevin A. . August 2003 . Adic Topologies for the Rational Integers . Canadian Journal of Mathematics . en . 55 . 4 . 711–723 . 10.4153/CJM-2003-030-3 . 121286344 . 0008-414X . free .
- Szczuka . Paulina . The Connectedness of Arithmetic Progressions in Furstenberg's, Golomb's, and Kirch's Topologies . 2010-10-01 . Demonstratio Mathematica . en . 43 . 4 . 899–910 . 10.1515/dema-2010-0416 . 122415499 . 2391-4661. free .
- Clark . Pete L. . Lebowitz-Lockard . Noah . Pollack . Paul . 2018-02-23 . A note on Golomb topologies . Quaestiones Mathematicae . 42 . 1 . 73–86 . 10.2989/16073606.2018.1438533 . 126371036 . 1607-3606.
- Spirito . Dario . 2019-06-24 . The Golomb topology on a Dedekind domain and the group of units of its quotients . math.GN . 1906.09922 .
- Spirito . Dario . 2019-11-06 . The Golomb topology of polynomial rings . math.GN . 1911.02328 .