proximity space

Let X be a set. A binary relationMathworldPlanetmath δ on P(X), the power setMathworldPlanetmath of X, is called a nearness relation on X if it satisfies the following conditions: for A,BP(X),

  1. 1.

    if AB, then AδB;

  2. 2.

    if AδB, then A and B;

  3. 3.

    (symmetry) if AδB, then BδA;

  4. 4.

    (A1A2)δB iff A1δB or A2δB;

  5. 5.

    AδB implies the existence of CX with AδC and (X-C)δB, where AδB means (A,B)δ.

If x,yX and AX, we write xδA to mean {x}δA, and xδy to mean {x}δ{y}.

When AδB, we say that A is δ-near, or just near B. δ is also called a proximity relation, or proximity for short. Condition 1 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying if AδB, then AB=. Condition 4 says that if A is near B, then any supersetMathworldPlanetmath of A is near B. Conversely, if A is not near B, then no subset of A is near B. In particular, if xA and AδB, then xδB.

Definition. A set X with a proximity as defined above is called a proximity space.

For any subset A of X, define Ac={xXxδA}. Then c is a closure operatorPlanetmathPlanetmathPlanetmath on X:


Clearly c=. Also AAc for any AX. To see Acc=Ac, assume xδAc, we want to show that xδA. If not, then there is CX such that xδC and (X-C)δA. The second part says that if yX-C, then yδA, which is equivalent to AcC. But xδC, so xδAc. Finally, x(AB)c iff xδ(AB) iff xδA or xδB iff xAc or xBc.∎

This turns X into a topological spaceMathworldPlanetmath. Thus any proximity space is a topological space induced by the closure operator defined above.

A proximity space is said to be separated if for any x,yX, xδy implies x=y.


  • Let (X,d) be a pseudometric space. For any xX and AX, define d(x,A):=infyAd(x,y). Next, for BX, define d(A,B):=infxAd(x,B). Finally, define AδB iff d(A,B)=0. Then δ is a proximity and (X,d) is a proximity space as a result.

  • discrete proximity. Let X be a non-empty set. For A,BX, define AδB iff AB. Then δ so defined is a proximity on X, and is called the discrete proximity on X.

  • indiscrete proximity. Again, X is a non-empty set and A,BX. Define AδB iff A and B. Then δ is also a proximity. It is called the indiscrete proximity on X.


  • 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
  • 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
Title proximity space
Canonical name ProximitySpace
Date of creation 2013-03-22 16:48:11
Last modified on 2013-03-22 16:48:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Definition
Classification msc 54E05
Synonym near
Synonym proximity
Synonym proximity relation
Defines nearness relation
Defines separated proximity space
Defines discrete proximity
Defines indiscrete proximity