PlanetMath: star refinement

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star refinement

(Definition)

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

$\displaystyle \star(A,\mathscr{C}):=\bigcup \lbrace C_i\in \mathscr{C} \mid C_i\cap A\neq \varnothing \rbrace.$

When

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

Properties of $\star$

$A\subseteq \star(A,\mathscr{C})$ .
If $A\subseteq B$ , then $\star(B,\mathscr{C})\subseteq \star(A,\mathscr{C})$ .
For any cover $\mathscr{C}$ of , 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 .
$\mathscr{C}\preceq \mathscr{C}^{\star} \preceq \mathscr{C}^b$ ( $\preceq$ denotes cover refinement).

Definitions. Let $\mathscr{C},\mathscr{D}$ be two covers of . 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 $\mathscr{C} \preceq^b\mathscr{D}\Rightarrow \mathscr{C} \preceq^{\star}\mathscr{D}\Rightarrow \mathscr{C} \preceq \mathscr{D}$ .

Bibliography

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

"star refinement" is owned by CWoo.

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Also defines:

star, star refine, barycentric refinement

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Cross-references: easy to see, definitions, refinement, properties, singleton, universe, subsets, cover
There are 13 references to this entry.

This is version 3 of star refinement, born on 2007-02-23, modified 2007-05-26.
Object id is 8959, canonical name is StarRefinement.
Accessed 1244 times total.

Classification:

AMS MSC:

54A99 (General topology :: Generalities :: Miscellaneous)

Pending Errata and Addenda

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