star refinement
StarRefinement
2007-02-23 14:17:02
2009-01-01 13:29:42
Definition
cover
star
star refine
barycentric refinement
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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 \emph{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 $\star(x,\mathscr{C})=\star(\lbrace x\rbrace, \mathscr{C})$.
\textbf{Properties of $\star$}
\begin{enumerate}
\item $A\subseteq \star(A,\mathscr{C})$.
\item If $A\subseteq B$, then $\star(A,\mathscr{C})\subseteq \star(B,\mathscr{C})$.
\item 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 $X$.
\item $\mathscr{C}\preceq \mathscr{C}^b \preceq \mathscr{C}^{\star}$ ($\preceq$ denotes cover refinement).
\end{enumerate}
\textbf{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 \emph{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 \emph{barycentric refinement} of $\mathscr{D}$, denoted by $\mathscr{C} \preceq^b \mathscr{D}$.
\textbf{Remark}. By property 4 above, it is easy to see that
$\mathscr{C} \preceq^{\star}\mathscr{D}\Rightarrow \mathscr{C} \preceq^b\mathscr{D}\Rightarrow \mathscr{C} \preceq \mathscr{D}$.
\begin{thebibliography}{9}
\bibitem{willard} S. Willard, \emph{General Topology},
Addison-Wesley, Publishing Company, 1970.
\end{thebibliography}