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 ...

, a branch of mathematics, the Appert topology, named for , is a

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

on the set of

positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space ''X'' with the Appert topology is called the Appert space.

Construction

For a subset ''S'' of ''X'', let denote the number of elements of ''S'' which are less than or equal to ''n'': :

\mathrm(n,S) = \#\ .

''S'' is defined to be open in the Appert topology if either it does not contain 1 or if it has

asymptotic density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the des ...

equal to 1, i.e., it satisfies :

\lim_ \frac = 1

. The empty set is open because it does not contain 1, and the whole set ''X'' is open since

\text(n,X)/n=1

for all ''n''.

Related topologies

The Appert topology is closely related to the

Fort space In mathematics, there are a few topological spaces named after M. K. Fort, Jr. Fort space Fort space is defined by taking an infinite set ''X'', with a particular point ''p'' in ''X'', and declaring open the subsets ''A'' of ''X'' such that: * ...

topology that arises from giving the set of integers greater than one the

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

, and then taking the point 1 as the point at infinity in a one point compactification of the space. The Appert topology is finer than the Fort space topology, as any cofinite subset of ''X'' has asymptotic density equal to 1.

Properties

* The closed subsets ''S'' of ''X'' are those that either contain 1 or that have zero asymptotic density, namely

\lim_ \mathrm(n,S)/n = 0

. * Every point of ''X'' has a

local basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...

clopen sets 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 d ...

, i.e., ''X'' is a

zero-dimensional space In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical i ...

.
''Proof'': Every open neighborhood of 1 is also closed. For any

x\ne 1

\

is both closed and open. * ''X'' is Hausdorff and

perfectly normal ''Perfectly Normal'' is a Canadian comedy film directed by Yves Simoneau, which premiered at the 1990 Festival of Festivals, before going into general theatrical release in 1991. Simoneau's first English-language film, it was written by Eugene Lip ...

(T₆).
''Proof'': ''X'' is T₁. Given any two disjoint closed sets ''A'' and ''B'', at least one of them, say ''A'', does not contain 1. ''A'' is then clopen and ''A'' and its complement are disjoint respective neighborhoods of ''A'' and ''B'', which shows that ''X'' is normal and Hausdorff. Finally, any subset, in particular any closed subset, in a countable T₁ space is a G_δ, so ''X'' is perfectly normal. * ''X'' is countable, but not

first countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...

, and hence not

second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...

and not

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 , \infty) ...

. * A subset of ''X'' is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

if and only if it is finite. In particular, ''X'' is not

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

, since there is no compact neighborhood of 1. * ''X'' is not

countably compact In mathematics a topological space is called countably compact if every countable open cover has a finite subcover. Equivalent definitions A topological space ''X'' is called countably compact if it satisfies any of the following equivalent conditi ...

.
''Proof:'' The infinite set

\

has zero asymptotic density, hence is closed in ''X''. Each of its points is isolated. Since ''X'' contains an infinite closed discrete subset, it is not

limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...

, and therefore it is not countably compact.

Notes

References

* . * {{Citation, first=L. A., last=Steen, first2=J. A., last2=Seebach, title=

Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) hav ...

, publisher=Dover, year=1995, ISBN=0-486-68735-X. General topology Topological spaces

Construction

Related topologies

Properties

See also

Notes

References