mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, if

\kappa

is a

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

uncountable In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...

cardinal Cardinal or The Cardinal most commonly refers to * Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of three species in the family Cardinalidae ***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...

then

\operatorname(\kappa),

the

filter Filtration is a physical process that separates solid matter and fluid from a mixture. Filter, filtering, filters or filtration may also refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Fil ...

of all sets containing a club subset of

\kappa,

is a

\kappa

-complete filter closed under

diagonal intersection Diagonal intersection is a term used in mathematics, especially in set theory. If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha<\delta\rangle is a

called the club filter. To see that this is a filter, note that

\kappa \in \operatorname(\kappa)

since it is thus both closed and unbounded (see

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 o ...

). If

x\in\operatorname(\kappa)

then any

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

\kappa

containing

x

is also in

\operatorname(\kappa),

since

x,

and therefore anything containing it, contains a club set. It is a

\kappa

-complete filter because the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of fewer than

\kappa

club sets is a club set. To see this, suppose

\langle C_i\rangle_

is a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of club sets where

\alpha < \kappa.

Obviously

C = \bigcap C_i

is closed, since any sequence which appears in

C

appears in every

C_i,

and therefore its limit is also in every

C_i.

To show that it is unbounded, take some

\beta < \kappa.

Let

\langle \beta_\rangle

be an increasing sequence with

\beta_ > \beta

and

\beta_ \in C_i

for every

i < \alpha.

Such a sequence can be constructed, since every

C_i

is unbounded. Since

\alpha < \kappa

and

\kappa

is regular, the limit of this sequence is less than

\kappa.

We call it

\beta_2,

and define a new sequence

\langle\beta_\rangle

similar to the previous sequence. We can repeat this process, getting a sequence of sequences

\langle\beta_\rangle

where each element of a sequence is greater than every member of the previous sequences. Then for each

i < \alpha,

\langle\beta_\rangle

is an increasing sequence contained in

C_i,

and all these sequences have the same limit (the limit of

\langle\beta_\rangle

). This limit is then contained in every

C_i,

and therefore

C,

and is greater than

\beta.

To see that

\operatorname(\kappa)

is closed under diagonal intersection, let

\langle C_i\rangle,

i < \kappa

be a sequence of club sets, and let

C = \Delta_ C_i.

To show

C

is closed, suppose

S\subseteq \alpha < \kappa

and

\bigcup S = \alpha.

Then for each

\gamma \in S,

\gamma \in C_\beta

for all

\beta < \gamma.

Since each

C_\beta

is closed,

\alpha \in C_\beta

for all

\beta < \alpha,

\alpha \in C.

To show

C

is unbounded, let

\alpha < \kappa,

and define a sequence

\xi_i,

i < \omega

as follows:

\xi_0 = \alpha,

and

\xi_

is the minimal element of

\bigcap_ C_\gamma

such that

\xi_ > \xi_i.

Such an element exists since by the above, the intersection of

\xi_i

club sets is club. Then

\xi = \bigcup_ \xi_i > \alpha

and

\xi \in C,

since it is in each

C_i

with

i < \xi.

References

* Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . {{Set theory Set theory

See also

References