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general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, branches of mathematics, one can define various topologies on the set \mathbb of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
or the set \mathbb_ of positive integers by taking as a base a suitable collection of
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s, sequences of the form \ or \. The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on \mathbb, and the Golomb topology and the Kirch topology on \mathbb_. 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 and provides an example of a
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
that is connected. The third topology, introduced by A.M. Kirch, is an example of a countably infinite Hausdorff space that is both connected and
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedne ...
. These topologies also have interesting
separation Separation may refer to: Films * ''Separation'' (1967 film), a British feature film written by and starring Jane Arden and directed by Jack Bond * ''La Séparation'', 1994 French film * ''A Separation'', 2011 Iranian film * ''Separation'' (20 ...
and
homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
properties. The notion of an arithmetic progression topology can be generalized to arbitrary
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
s.


Construction

Two-sided arithmetic progressions in \mathbb are subsets of the form : a\mathbb+b := \, where a,b\in\mathbb and a>0. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form: : (a\mathbb+b) \cap (c\mathbb+b) = \operatorname(a,c)\mathbb+b, where \operatorname(a,c) is the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
of a and c. Similarly, one-sided arithmetic progressions in \mathbb_=\ are subsets of the form : a\mathbb+b := \ = \, with \mathbb=\ and a,b>0. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form: : (a\mathbb+b) \cap (c\mathbb+d) = \operatorname(a,c)\mathbb+q, with q 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 \mathbb or \mathbb_ by choosing a collection \mathcal of arithmetic progressions, declaring all elements of \mathcal to be open sets, and taking the topology generated by those. If any nonempty intersection of two elements of \mathcal is again an element of \mathcal, the collection \mathcal will be a base for the topology. In general, it will be a
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by s ...
for the topology, and the set of all arithmetic progressions that are nonempty finite intersections of elements of \mathcal will be a base for the topology. Three special cases follow. The Furstenberg topology, or evenly spaced integer topology,Steen & Seebach, pp. 80-81, counterexample #58 on the set \mathbb of integers is obtained by taking as a base the collection of all a\mathbb+b with a,b\in\mathbb and a>0. The Golomb topology, or relatively prime integer topology,Steen & Seebach, pp. 82-84, counterexample #60 on the set \mathbb_ of positive integers is obtained by taking as a base the collection of all a\mathbb+b with a,b>0 and a and b
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
. Equivalently, the subcollection of such sets with the extra condition b also forms a base for the topology. The corresponding
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is called the Golomb space. The Kirch topology, or prime integer topology,Steen & Seebach, pp. 82-84, counterexample #61 on the set \mathbb_ of positive integers is obtained by taking as a ''subbase'' the collection of all p\mathbb+b with b>0 and p prime not dividing b. Equivalently, one can take as a subbase the collection of all p\mathbb+b with p prime and 0. A ''base'' for the topology consists of all a\mathbb+b with relatively prime a,b>0 and a squarefree (or the same with the additional condition b). The corresponding topological space is called the 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 \mathbb_). On the set \mathbb_, 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 The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
. The Furstenberg topology is Hausdorff and regular. It is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
, but not completely metrizable. Indeed, it is homeomorphic to the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s \mathbb with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
inherited from the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. Broughan 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 In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
. The Golomb topology is connected, but not
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedne ...
. The Kirch topology is both connected and locally connected. The integers with the Furstenberg topology form a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ...
, because it is a topological ring — in some sense, the only topology on \mathbb for which it is a ring. By contrast, the Golomb space and the Kirch space are topologically rigid — the only self-
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
is the trivial one.


Relation to the infinitude of primes

Both the Furstenberg and Golomb topologies furnish a proof that there are infinitely many
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s. 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 residue classes modulo . Each residue class is an arithmetic progression, and thus
clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
. # 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 \Z\subset \hat\Z, where \hat\Z is the
profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mat ...
ring with its profinite topology. The notion of an arithmetic progression makes sense in arbitrary \mathbb- modules, but the construction of a topology on them relies on closure under intersection. Instead, the correct generalization builds a topology out of
ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
of a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
. 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.


Notes


References

* . * {{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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...
, location=Berlin, New York , edition=
Dover Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone ...
reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995 Topological spaces Arithmetic