Arithmetic progression topologies

Two-sided arithmetic progressions in $\backslash mathbb\{Z\}$ are subsets of the form

: $a\backslash mathbb\{Z\}+b\; :=\; \backslash \{an+b\; :\; n\backslash in\backslash mathbb\{Z\}\backslash \},$

where $a,b\backslash in\backslash mathbb\{Z\}$ and $a>0.$ The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:

: $(a\backslash mathbb\{Z\}+b)\; \backslash cap\; (c\backslash mathbb\{Z\}+b)\; =\; \backslash operatorname\{lcm\}(a,c)\backslash mathbb\{Z\}+b,$

where $\backslash operatorname\{lcm\}(a,c)$ is the least common multiple of $a$ and $c.$Steen & Seebach, p. 82, counterexample #60, item 1

Similarly, one-sided arithmetic progressions in $\backslash mathbb\{Z\}\_\{>0\}=\backslash \{1,2,...\backslash \}$ are subsets of the form

: $a\backslash mathbb\{N\}+b\; :=\; \backslash \{an+b\; :\; n\backslash in\backslash mathbb\{N\}\backslash \}\; =\; \backslash \{b,a+b,2a+b,...\backslash \},$

with $\backslash mathbb\{N\}=\backslash \{0,1,2,...\backslash \}$ and $a,b>0$. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:

: $(a\backslash mathbb\{N\}+b)\; \backslash cap\; (c\backslash mathbb\{N\}+d)\; =\; \backslash operatorname\{lcm\}(a,c)\backslash mathbb\{N\}+q,$

with $q$ equal to the smallest element in the intersection.

{{anchor|Similar}}

{{anchor|Definitions}}

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 $\backslash mathbb\{Z\}$ or $\backslash mathbb\{Z\}\_\{>0\}$ by choosing a collection $\backslash mathcal\{B\}$ of arithmetic progressions, declaring all elements of $\backslash mathcal\{B\}$ to be open sets, and taking the topology generated by those. If any nonempty intersection of two elements of $\backslash mathcal\{B\}$ is again an element of $\backslash mathcal\{B\}$, the collection $\backslash mathcal\{B\}$ 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 $\backslash mathcal\{B\}$ will be a base for the topology. Three special cases follow.

The Furstenberg topology,{{sfn|Furstenberg|1955}} or evenly spaced integer topology,Steen & Seebach, pp. 80-81, counterexample #58 on the set $\backslash mathbb\{Z\}$ of integers is obtained by taking as a base the collection of all $a\backslash mathbb\{Z\}+b$ with $a,b\backslash in\backslash mathbb\{Z\}$ and $a>0.$

The Golomb topology, or relatively prime integer topology,Steen & Seebach, pp. 82-84, counterexample #60 on the set $\backslash mathbb\{Z\}\_\{>0\}$ of positive integers is obtained by taking as a base the collection of all $a\backslash mathbb\{N\}+b$ with $a,b>0$ and $a$ and $b$ relatively prime. Equivalently,{{cite web |title=The Kirch topology is the same as the prime integer topology |url=https://math.stackexchange.com/questions/4460847}} the subcollection of such sets with the extra condition Golomb space.{{Cite journal |last1=Banakh |first1=Taras |last2=Spirito |first2=Dario |last3=Turek |first3=Sławomir |date=2021-10-28 |title=The Golomb space is topologically rigid |arxiv=1912.01994 |journal=Commentationes Mathematicae Universitatis Carolinae |volume=62 |issue=3 |pages=347–360 |doi=10.14712/1213-7243.2021.023 |s2cid=240183836 |issn=0010-2628}}

The Kirch topology, or prime integer topology,Steen & Seebach, pp. 82-84, counterexample #61 on the set $\backslash mathbb\{Z\}\_\{>0\}$ of positive integers is obtained by taking as a subbase the collection of all $p\backslash mathbb\{N\}+b$ with $b>0$ and $p$ prime not dividing $b.${{Cite journal |last1=Banakh |first1=Taras |last2=Stelmakh |first2=Yaryna |last3=Turek |first3=Sławomir |date=2021-12-01 |title=The Kirch space is topologically rigid |arxiv=2006.12357 |journal=Topology and Its Applications |volume=304 |page=107782 |doi=10.1016/j.topol.2021.107782|s2cid=219966624 }}

Equivalently, one can take as a subbase the collection of all $p\backslash mathbb\{N\}+b$ with $p$ prime and base for the topology consists of all $a\backslash mathbb\{N\}+b$ with relatively prime $a,b>0$ and $a$ squarefree (or the same with the additional condition Kirch space.

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 $\backslash mathbb\{Z\}\_\{>0\}$). On the set $\backslash mathbb\{Z\}\_\{>0\}$, the Kirch topology is coarser than the Golomb topology, which is itself coarser that the Furstenberg topology.

The Golomb topology and the Kirch topology are Hausdorff, but not regular.

The Furstenberg topology is Hausdorff and regular. It is metrizable, but not completely metrizable.{{Cite journal |last1 = Lovas | first1 = R. | last2 = Mező | first2 = I. |title = Some observations on the Furstenberg topological space |journal = Elemente der Mathematik |volume = 70 |pages = 103–116 |year = 2015 |issue = 3 |doi = 10.4171/EM/283 | s2cid = 126337479 |url = http://www.ems-ph.org/doi/10.4171/EM/283}} Indeed, it is homeomorphic to the rational numbers $\backslash mathbb\{Q\}$ with the subspace topology inherited from the real line.{{Cite journal |last=Broughan |first=Kevin A. |date=August 2003 |title=Adic Topologies for the Rational Integers |journal=Canadian Journal of Mathematics |language=en |volume=55 |issue=4 |pages=711–723 |doi=10.4153/CJM-2003-030-3 |s2cid=121286344 |issn=0008-414X |doi-access=free }} Broughan has shown that the Furstenberg topology is closely related to the P-adic number of the rational numbers.

Regarding connectedness properties, the Furstenberg topology is totally disconnected. The Golomb topology is connected,{{Cite journal |last=Szczuka |first=Paulina |title=The Connectedness of Arithmetic Progressions in Furstenberg's, Golomb's, and Kirch's Topologies |date=2010-10-01 |journal=Demonstratio Mathematica |language=en |volume=43 |issue=4 |pages=899–910 |doi=10.1515/dema-2010-0416 |s2cid=122415499 |issn=2391-4661|doi-access=free }} but not locally connected.{{sfn|Kirch|1969|ps=, Theorem 1}} The Kirch topology is both connected and locally connected.

The integers with the Furstenberg topology form a homogeneous space, because it is a topological ring — in some sense, the only topology on $\backslash mathbb\{Z\}$ for which it is a ring.{{sfn|Broughan|2003|ps=, Theorem 2.1}} By contrast, the Golomb space and the Kirch space are topologically rigid — the only self-homeomorphism is the trivial one.

{{Main|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.{{sfn|Furstenberg|1955}} A sketch of the proof runs as follows:

1. Fix a prime {{mvar|p}} and note that the (positive, in the Golomb space case) integers are a union of finitely many residue classes modulo {{mvar|p}}. Each residue class is an arithmetic progression, and thus clopen.
2. 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 {{math|±1}}.
3. If there are finitely many primes, that union is a closed set, and so its complement ({{math|{±1}}}) is open.
4. But every nonempty open set is infinite, so {{math|{±1}}} is not open.

The Furstenberg topology is a special case of the profinite topology on a group. In detail, it is the topology induced by the inclusion $\backslash Z\backslash subset\; \backslash hat\backslash Z$, where $\backslash hat\backslash Z$ is the profinite integer ring with its profinite topology.

The notion of an arithmetic progression makes sense in arbitrary $\backslash mathbb\{Z\}$-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.{{Cite journal |last1=Clark |first1=Pete L. |last2=Lebowitz-Lockard |first2=Noah |last3=Pollack |first3=Paul |date=2018-02-23 |title=A note on Golomb topologies |url=https://www.tandfonline.com/doi/pdf/10.2989/16073606.2018.1438533 |journal=Quaestiones Mathematicae |volume=42 |issue=1 |pages=73–86 |doi=10.2989/16073606.2018.1438533 |s2cid=126371036 |issn=1607-3606}} 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.{{cite arXiv |last=Spirito |first=Dario |date=2019-06-24 |title=The Golomb topology on a Dedekind domain and the group of units of its quotients |class=math.GN |eprint=1906.09922 }}{{cite arXiv |last=Spirito |first=Dario |date=2019-11-06 |title=The Golomb topology of polynomial rings |class=math.GN |eprint=1911.02328 }}

{{Reflist}}

• {{citation |last=Furstenberg |first=Harry |title=On the infinitude of primes |journal=American Mathematical Monthly |volume=62 |issue=5 |pages=353 |year=1955 |publisher=Mathematical Association of America |doi=10.2307/2307043 |jstor=2307043 |mr=0068566 |authorlink=Hillel Furstenberg}}.
• {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | title-link=Counterexamples in Topology | orig-year=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 }}

Category:Topological spaces

Category:Arithmetic series