In
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, geometri ...
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, "Mat ...
, branches of
mathematics, one can define various
topologies
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
on the set
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
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
, and the Golomb topology and the Kirch topology on
. Precise definitions are given below.
Hillel Furstenberg
Hillel (Harry) Furstenberg ( he, הלל (הארי) פורסטנברג) (born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy o ...
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
Solomon (; , ),, ; ar, سُلَيْمَان, ', , ; el, Σολομών, ; la, Salomon also called Jedidiah ( Hebrew: , Modern: , Tiberian: ''Yăḏīḏăyāh'', "beloved of Yah"), was a monarch of ancient Israel and the son and succe ...
and provides an example of a
countably infinite 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 m ...
that is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. 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, connectedness ...
. These topologies also have interesting
separation and
homogeneity
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform in compos ...
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 necessarily ...
s.
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:
:
where
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
and
Similarly, one-sided arithmetic progressions in
are subsets of the form
:
with
and
. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:
:
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
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 so ...
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,
[Steen & Seebach, pp. 80-81, counterexample #58] on the set
of integers is obtained by taking as a base the collection of all
with
and
The Golomb topology,
or relatively prime integer topology,
[Steen & Seebach, pp. 82-84, counterexample #60] on the set
of positive integers is obtained by taking as a base the collection of all
with
and
and
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 equivale ...
.
Equivalently,
the subcollection of such sets with the extra condition