star refinement

Let X be a set and 𝒞={Ci∣i∈I} be a cover of X (we assume Ci and X are all subsets of some universe). Let A⊆X. 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 A⊆B, then ⋆(A,𝒞)⊆⋆(B,𝒞).

  3. 3.

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

  4. 4.

    𝒞âȘŻđ’žbâȘŻđ’žâ‹† (âȘŻ denotes cover refinement).

Definitions. Let 𝒞,𝒟 be two covers of X. If 𝒞⋆âȘŻđ’Ÿ, then we say that 𝒞 is a star refinement of 𝒟, denoted by 𝒞âȘŻâ‹†đ’Ÿ. If 𝒞bâȘŻđ’Ÿ, then we say that 𝒞 is a barycentric refinement of 𝒟, denoted by 𝒞âȘŻb𝒟.

Remark. By property 4 above, it is easy to see that 𝒞âȘŻâ‹†đ’Ÿâ‡’đ’žâȘŻb𝒟⇒𝒞âȘŻđ’Ÿ.


  • 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