star refinement

Let X be a set and 𝒞={CiiI} be a cover of X (we assume Ci and X are all subsets of some universe). Let AX. The star of A (with respect to the cover 𝒞) is defined as


When A is a singleton, we write (x,𝒞)=({x},𝒞).

Properties of

  1. 1.


  2. 2.

    If AB, then (A,𝒞)(B,𝒞).

  3. 3.

    For any cover 𝒞 of X, the sets 𝒞:={(Ci,𝒞)Ci𝒞} and 𝒞b:={(x,𝒞)xX} are both covers of X.

  4. 4.

    𝒞𝒞b𝒞 ( denotes cover refinement).

Definitions. Let 𝒞,𝒟 be two covers of X. If 𝒞𝒟, then we say that 𝒞 is a star refinement of 𝒟, denoted by 𝒞𝒟. If 𝒞b𝒟, then we say that 𝒞 is a barycentric refinement of 𝒟, denoted by 𝒞b𝒟.

Remark. By property 4 above, it is easy to see that 𝒞𝒟𝒞b𝒟𝒞𝒟.


  • 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title star refinement
Canonical name StarRefinement
Date of creation 2013-03-22 16:44:13
Last modified on 2013-03-22 16:44:13
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 54A99
Defines star
Defines star refine
Defines barycentric refinement