closure space

Call a set X with a closure operatorPlanetmathPlanetmathPlanetmath defined on it a closure space.

Every topological spaceMathworldPlanetmath is a closure space, if we define the closure operator of the space as a function that takes any subset to its closureMathworldPlanetmath. The converseMathworldPlanetmath is also true:

Proposition 1.

Let X be a closure space with c the associated closure operator. Define a “closed setPlanetmathPlanetmath” of X as a subset A of X such that Ac=A, and an “open set” of X as the complementPlanetmathPlanetmath of some closed set of X. Then the collectionMathworldPlanetmath T of all open sets of X is a topology on X.


Since c=, is closed. Also, XXc and XcX imply that Xc=X, or X is closed. If A,BX are closed, then (AB)c=AcBc=AB is closed as well. Finally, suppose Ai are closed. Let B=Ai. For each i, Ai=BAi, so Ai=Aic=(BAi)c=BcAic=BcAi. This means BcAi, or BcAi=B. But BBc by definition, so B=Bc, or that Ai is closed. ∎

𝒯 so defined is called the closure topology of X with respect to the closure operator c.


  1. 1.

    A closure space can be more generally defined as a set X together with an operator cl:P(X)P(X) such that cl satisfies all of the Kuratowski’s closure axioms where the equal sign “=” is replaced with set inclusion”, and the preservation of is no longer assumed.

  2. 2.

    Even more generally, a closure space can be defined as a set X and an operator cl on P(X) such that

    • Acl(A),

    • cl(cl(A))cl(A), and

    • cl is order-preserving, i.e., if AB, then cl(A)cl(B).

    It can be easily deduced that cl(A)cl(B)cl(AB). In general however, the equality fails. The three axioms above can be shown to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to a single axiom:

    Acl(B) iff cl(A)cl(B).
  3. 3.

    In a closure space X, a subset A of X is said to be closed if cl(A)=A. Let C(X) be the set of all closed sets of X. It is not hard to see that if C(X) is closed under , then cldistributes over, that is, we have the equality cl(A)cl(B)=cl(AB).

  4. 4.

    Also, cl() is the smallest closed set in X; it is the bottom element in C(X). This means that if there are two disjoint closed sets in X, then cl()=. This is equivalent to saying that is closed whenever there exist A,BX such that cl(A)cl(B)=.

  5. 5.

    Since the distributivity of cl over does not hold in general, and there is no guarantee that cl()=, a closure space under these generalized versions is a more general system than a topological space.


  • 1 N. M. Martin, S. Pollard: Closure Spaces and Logic, Springer, (1996).
Title closure space
Canonical name ClosureSpace
Date of creation 2013-03-22 16:48:08
Last modified on 2013-03-22 16:48:08
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Derivation
Classification msc 54A05
Defines closure topology