PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] star refinement (Definition)

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.




"star refinement" is owned by CWoo.
(view preamble | get metadata)

View style:

Also defines:  star, star refine, barycentric refinement

This object's parent.
Log in to rate this entry.
(view current ratings)

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

This is version 4 of star refinement, born on 2007-02-23, modified 2009-01-01.
Object id is 8959, canonical name is StarRefinement.
Accessed 2666 times total.

Classification:
AMS MSC54A99 (General topology :: Generalities :: Miscellaneous)

Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)