Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Formal definitions

Formally, if

κ

{\displaystyle \kappa }

is a limit ordinal, then a set

C

⊆

κ

{\displaystyle C\subseteq \kappa }

is closed in

κ

{\displaystyle \kappa }

if and only if for every

α

<

κ

,

{\displaystyle \alpha <\kappa ,}

if

sup

(

C

∩

α

)

=

α

≠

0

,

{\displaystyle \sup(C\cap \alpha )=\alpha \neq 0,}

then

α

∈

C

.

{\displaystyle \alpha \in C.}

Thus, if the limit of some sequence from

C

{\displaystyle C}

is less than

κ

,

{\displaystyle \kappa ,}

then the limit is also in

C

.

{\displaystyle C.}

If

κ

{\displaystyle \kappa }

is a limit ordinal and

C

⊆

κ

{\displaystyle C\subseteq \kappa }

then

C

{\displaystyle C}

is unbounded in

κ

{\displaystyle \kappa }

if for any

α

<

κ

,

{\displaystyle \alpha <\kappa ,}

there is some

β

∈

C

{\displaystyle \beta \in C}

such that

α

<

β

.

{\displaystyle \alpha <\beta .}

If a set is both closed and unbounded, then it is a club set.

Closed proper classes of ordinals are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

Examples

The set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.

If

κ

{\displaystyle \kappa }

is an uncountable initial ordinal, then the set of all limit ordinals

α

<

κ

{\displaystyle \alpha <\kappa }

is closed unbounded in

κ

.

{\displaystyle \kappa .}

In fact a club set is nothing else but the range of a normal function (i.e.