In
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 ...
, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize
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 points ...
s.
The concept was described by but ignored at the time.
[W. J. Thron, ''Frederic Riesz' contributions to the foundations of general topology'', in C.E. Aull and R. Lowen (eds.), ''Handbook of the History of General Topology'', Volume 1, 21-29, Kluwer 1997.] It was rediscovered and axiomatized by
V. A. Efremovič in 1934 under the name of infinitesimal space, but not published until 1951. In the interim, discovered a version of the same concept under the name of separation space.
Definition
A
is a set
with a
relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
between subsets of
satisfying the following properties:
For all subsets
#
implies
#
implies
#
implies
#
implies (
or
)
# (For all
or
) implies
Proximity without the first axiom is called (but then Axioms 2 and 4 must be stated in a two-sided fashion).
If
we say
is near
or
and
are ; otherwise we say
and
are . We say
is a or of
written
if and only if
and
are apart.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets
#
#
implies
#
implies
# (
and
) implies
#
implies
#
implies that there exists some
such that
A proximity space is called if
implies
A or is one that preserves nearness, that is, given
if
in
then