| Version 3 |
Version 2 |
|
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
|
Let $X$ be 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
|
| $$\star(A,\mathscr{C}):=\bigcup \lbrace C_i\in \mathscr{C} \mid C_i\cap A\neq \varnothing \rbrace.$$ |
$$\star(A,\mathscr{C}):=\bigcup \lbrace C_i\in \mathscr{C} \mid C_i\cap A\neq \varnothing \rbrace.$$ |
| When $A$ is a singleton, we write $\star(x,\mathscr{C})=\star(\lbrace x\rbrace, \mathscr{C})$. |
When $A$ is a singleton, we write $\star(x,\mathscr{C})=\star(\lbrace x\rbrace, \mathscr{C})$. |
|
|
| \textbf{Properties of $\star$} |
\textbf{Properties of $\star$} |
| \begin{enumerate} |
\begin{enumerate} |
| \item $A\subseteq \star(A,\mathscr{C})$. |
\item $A\subseteq \star(A,\mathscr{C})$. |
| \item If $A\subseteq B$, then $\star(B,\mathscr{C})\subseteq \star(A,\mathscr{C})$. |
\item If $A\subseteq B$, then $\star(B,\mathscr{C})\subseteq \star(A,\mathscr{C})$. |
| \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 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 $\mathscr{C}\preceq \mathscr{C}^{\star} \preceq \mathscr{C}^b$ ($\preceq$ denotes cover refinement). |
\item $\mathscr{C}\preceq \mathscr{C}^{\star} \preceq \mathscr{C}^b$ ($\preceq$ denotes cover refinement). |
| \end{enumerate} |
\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 $\mathscr{C} \preceq^b \mathscr{D}$.
|
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 |
\textbf{Remark}. By property 4 above, it is easy to see that |
| $\mathscr{C} \preceq^b\mathscr{D}\Rightarrow \mathscr{C} \preceq^{\star}\mathscr{D}\Rightarrow \mathscr{C} \preceq \mathscr{D}$. |
$\mathscr{C} \preceq^b\mathscr{D}\Rightarrow \mathscr{C} \preceq^{\star}\mathscr{D}\Rightarrow \mathscr{C} \preceq \mathscr{D}$. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{willard} S. Willard, \emph{General Topology}, |
\bibitem{willard} S. Willard, \emph{General Topology}, |
| Addison-Wesley, Publishing Company, 1970. |
Addison-Wesley, Publishing Company, 1970. |
| \end{thebibliography} |
\end{thebibliography} |
