In mathematics, the Arens–Fort space is a special example in the theory of

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 ...

s, named for Richard Friederich Arens and M. K. Fort, Jr.

Definition

The Arens–Fort space is the topological space

(X,\tau)

where

X

is the set of ordered pairs of non-negative

integer 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 languag ...

(m, n).

A subset

U \subseteq X

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...

, that is, belongs to

\tau,

if and only if: *

U

does not contain

(0, 0),

or *

U

contains

(0, 0)

and also all but a finite number of points of all but a finite number of columns, where a column is a set

\

with

0 \leq m \in \mathbb

fixed. In other words, an open set is only "allowed" to contain

(0, 0)

if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points.

Properties

It is * Hausdorff * regular *

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

It is not: * second-countable *

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) ...

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) s ...

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 ...

There is no sequence in

X \setminus \

that converges to

(0, 0).

However, there is a sequence

x_ = \left( x_i \right)_^

X \setminus \

such that

(0, 0)

is a cluster point of

x_.

References

* {{DEFAULTSORT:Arens-Fort space Topological spaces

Definition

Properties

See also

References