partial ordering in a topological space

Let X be a topological spaceMathworldPlanetmath. For any x,yX, we define a binary relationMathworldPlanetmath on X as follows:

xy   iff   x{y}¯.
Proposition 1.

is a preorderMathworldPlanetmath.


Clearly xx. Next, suppose xy and yz. Let C be a closed setPlanetmathPlanetmath containing z. Since y is in the closureMathworldPlanetmathPlanetmath of {z}, yC. Since x is in the closure of {y}, xC also. So xz. ∎

We call the specialization preorder on X. If xy, then x is called a specialization point of y, and y a generization point of x. For any set AX,

  • the set of all specialization points of points of A is called the specialization of A, and is denoted by Sp(A);

  • the set of all generization points of points of A is called the generization of A, and is denoted by Gen(A).

Proposition 2.

. If X is T0 (, then is a partial orderMathworldPlanetmath.


Suppose next that xy and yx. If xy, then there is an open set A such that xA and yA. So yAc, the complementPlanetmathPlanetmath of A, which is a closed set. But then xAc since it is in the closure of {y}. So xAAc=, a contradition. Thus x=y. ∎

This turns a T0 topological space into a poset, where here is called the specialization order of the space.

Given a T0 space, we have the following:

Proposition 3.

xy iff xU implies yU for any open set U in X.


(): if xU and yU, then yUc. Since xy, we have xUc, a contradictionMathworldPlanetmathPlanetmath. (): if x{y}¯, then for some closed set C, we have yC and xC. But then xCc, so that yCc, a contradiction. ∎


  • {x}¯=x, the lower set of x. (zx iff zx iff z{x}¯).

  • But if X is T1 (, then the partial ordering just defined is trivial (the diagonal set), since every point is a closed point (for verification, just modify the antisymmetry portion of the above proof).

Title partial ordering in a topological space
Canonical name PartialOrderingInATopologicalSpace
Date of creation 2013-03-22 16:35:02
Last modified on 2013-03-22 16:35:02
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 54F99
Defines specialization order
Defines specialization preorder
Defines specialization
Defines generization