star refinement

Let $X$ be a set and $\mathscr{C}=\{C_{i}\mid i\in I\}$ 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 $\mathscr{C}$ ) is defined as

\star(A,\mathscr{C}):=\bigcup\{C_{i}\in\mathscr{C}\mid C_{i}\cap A\neq% \varnothing\}.

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

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 $\mathscr{C}^{\star}:=\{\star(C_{i},\mathscr{C})\mid C_{i}\in\mathscr{C}\}$ and $\mathscr{C}^{b}:=\{\star(x,\mathscr{C})\mid x\in X\}$ 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 $\mathscr{C}\preceq^{\star}\mathscr{D}\Rightarrow\mathscr{C}\preceq^{b}\mathscr% {D}\Rightarrow\mathscr{C}\preceq\mathscr{D}$ .

References

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