Let $X$ be a set and $\mathscr{C}=\lbrace C_i\mid i\in I\rbrace$ 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 $\mathscr{C}$ is defined as $$\star(A,\mathscr{C}):=\bigcup \lbrace C_i\in \mathscr{C} \mid C_i\cap A\neq \varnothing \rbrace.$$ When $A$ is a singleton, we write $\star(x,\mathscr{C})=\star(\lbrace x\rbrace, \mathscr{C})$

<</SPAN>#81#>Properties of $\star$

1. $A\subseteq \star(A,\mathscr{C})$
2. If $A\subseteq B$ then $\star(A,\mathscr{C})\subseteq \star(B,\mathscr{C})$
3. For any cover $\mathscr{C}$ of $X$ the sets $\mathscr{C}^{\star}:=\lbrace \star(C_i,\mathscr{C}) \mid C_i\in \mathscr{C}\rbrace$ and $\mathscr{C}^b:=\lbrace \star(x,\mathscr{C})\mid x\in X\rbrace$ are both covers of $X$
4. $\mathscr{C}\preceq \mathscr{C}^b \preceq \mathscr{C}^{\star}$ ($\preceq$ denotes cover refinement).

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

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

Bibliography

1

S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.