star refinement

Let $X$ be a set and $𝒞=\{C_i\mid i\in I\}$ be a cover of $X$ (we assume $C_i$ and $X$ are all subsets of some universe). Let $A\subseteq X$. The star of $A$ (with respect to the cover $𝒞$) is defined as

$$\star (A,𝒞):=\bigcup \{C_i\in 𝒞\mid C_i\cap A\ne \mathrm{\varnothing }\}.$$

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

Properties of $\mathrm{\star }$

1. $A\subseteq \star (A,𝒞)$.

2. If $A\subseteq B$, then $\star (A,𝒞)\subseteq \star (B,𝒞)$.

3. For any cover $𝒞$ of $X$, the sets ${𝒞}^{\star }:=\{\star (C_i,𝒞)\mid C_i\in 𝒞\}$ and ${𝒞}^b:=\{\star (x,𝒞)\mid x\in X\}$ are both covers of $X$.

4. $𝒞⪯{𝒞}^b⪯{𝒞}^{\star }$ ($⪯$ denotes cover refinement).

Definitions. Let $𝒞,𝒟$ be two covers of $X$. If ${𝒞}^{\star }⪯𝒟$, then we say that $𝒞$ is a star refinement of $𝒟$, denoted by $𝒞{⪯}^{\star }𝒟$. If ${𝒞}^b⪯𝒟$, then we say that $𝒞$ is a barycentric refinement of $𝒟$, denoted by $𝒞{⪯}^b𝒟$.

Remark. By property 4 above, it is easy to see that$𝒞{⪯}^{\star }𝒟\Rightarrow 𝒞{⪯}^b𝒟\Rightarrow 𝒞⪯𝒟$.