Club set explained
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 definition
Formally, if
is a limit ordinal, then a set
is
closed in
if and only if for every
if
\sup(C\cap\alpha)=\alpha ≠ 0,
then
Thus, if the
limit of some sequence from
is less than
then the limit is also in
If
is a limit ordinal and
then
is
unbounded in
if for any
there is some
such that
If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).
For example, 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
is an uncountable initial ordinal, then the set of all limit ordinals
is closed unbounded in
In fact a club set is nothing else but the range of a
normal function (i.e. increasing and continuous).
More generally, if
is a nonempty set and
is a
cardinal, then
(the set of subsets of
of cardinality
) is
club if every union of a subset of
is in
and every subset of
of cardinality less than
is contained in some element of
(see
stationary set).
The closed unbounded filter
See main article: Club filter. Let
be a limit ordinal of uncountable
cofinality
For some
, let
\langleC\xi:\xi<\alpha\rangle
be a sequence of closed unbounded subsets of
Then
is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any
and for each
n < ω choose from each
an element
which is possible because each is unbounded. Since this is a collection of fewer than
ordinals, all less than
their least upper bound must also be less than
so we can call it
This process generates a countable sequence
\beta0,\beta1,\beta2,\ldots.
The limit of this sequence must in fact also be the limit of the sequence
and since each
is closed and
is uncountable, this limit must be in each
and therefore this limit is an element of the intersection that is above
which shows that the intersection is unbounded. QED.
From this, it can be seen that if
is a
regular cardinal, then
\{S\subseteq\kappa:\existsC\subseteqSsuchthatCisclosedunboundedin\kappa\}
is a non-principal
-complete proper
filter on the set
(that is, on the
poset
).
If
is a regular cardinal then club sets are also closed under
diagonal intersection.
In fact, if
is regular and
is any filter on
closed under diagonal intersection, containing all sets of the form
\{\xi<\kappa:\xi\geq\alpha\}
for
then
must include all club sets.
References
- Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .
- Lévy, Azriel (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover.