Class 23 - Friday 10/09

Friday, October 9, 2015

12:08

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Stationary Sets.

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1. The union of fewer than K non-stationary subsets of K is non-stationary.

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2. If S C K is stationary in K and (Ai : i e I) is a partition of S into fewer than K subsets,
then Ai must be stationary for some i e I.

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The Non-stationary Ideal.

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Definition. Let X be a set and R P(X). Let a+b— (
a b) CJ (b x a) and a •b an b
for every a, b e R.

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Lemma. The set R with the operations of addition and multiplication defined as above is a
commutative ring with identity. A subset I C R is an ideal in R if and only if the following
conditions hold:
1.
2.
3.
if a, bel then alJbe I •
if a I and b C a, then be I.

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Definition. Let X be a set. A collection T of subsets of X is called an ideal on X iff it is a
proper ideal in the ring P(X) with operations as above.

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Lemma. Let K be a regular uncountable cardinal. The collection of all non-stationary subsets
of K is an ideal on K.