Stationary Set Article Index for
Stationary 		Website Links For
Stationary 		 

Information About

Stationary Set
  CATEGORIES ABOUT STATIONARY SET
   
model theory
   
set theory
   
ordinal numbers
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




CLASSICAL NOTION

If \kappa is a Cardinal of uncountable Cofinality , C\subseteq\kappa, and C Intersects every Club in \kappa, then C is called a stationary set. If C is not stationary then it is a '''thin set'''.

In fact the intersection of a stationary set and a club set is itself stationary. This is true because if S is stationary and C_1 , C_2 are club we have: S \cap (C_1 \cap C_2) = (S \cap C_1) \cap C_2. Now C_1 \cap C_2 is a club set as it is the intersection of two club sets. So S \cap (C_1 \cap C_2) is non empty. But then (S \cap C_1) must be stationary as C_2 is arbitrary.

''See also'': Fodor's Lemma

The restriction to uncountable cofinality is in order to avoid trivialities: Suppose \kappa has countable cofinality. Then S\subset\kappa is stationary in \kappa if and only if \kappa\setminus S is bounded in \kappa. In particular, if the cofinality of \kappa is \omega=\aleph_0, then any two stationary subsets of \kappa have stationary intersection.

This is no longer the case if the cofinality of \kappa is uncountable. In fact, suppose \kappa is Regular and S\subset\kappa is stationary. Then S can be partitioned into \kappa many disjoint stationary sets. This result is due to Solovay . If \kappa is a Successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix.


JECH'S NOTION