club filter

If κ is a regularPlanetmathPlanetmath uncountable cardinal then club(κ), the filter of all sets containing a club subset of κ, is a κ-completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath filter closed under diagonal intersection called the club filter.

To see that this is a filter, note that κclub(κ) since it is obviously both closed and unboundedPlanetmathPlanetmath. If xclub(κ) then any subset of κ containing x is also in club(κ), since x, and therefore anything containing it, contains a club set.

It is a κ complete filter because the intersectionMathworldPlanetmathPlanetmath of fewer than κ club sets is a club set. To see this, suppose Cii<α is a sequenceMathworldPlanetmathPlanetmath of club sets where α<κ. Obviously C=Ci is closed, since any sequence which appears in C appears in every Ci, and therefore its limit is also in every Ci. To show that it is unbounded, take some β<κ. Let β1,i be an increasing sequence with β1,1>β and β1,iCi for every i<α. Such a sequence can be constructed, since every Ci is unbounded. Since α<κ and κ is regular, the limit of this sequence is less than κ. We call it β2, and define a new sequence β2,i similar to the previous sequence. We can repeat this process, getting a sequence of sequences βj,i where each element of a sequence is greater than every member of the previous sequences. Then for each i<α, βj,i is an increasing sequence contained in Ci, and all these sequences have the same limit (the limit of βj,i). This limit is then contained in every Ci, and therefore C, and is greater than β.

To see that club(κ) is closed under diagonal intersection, let Ci, i<κ be a sequence, and let C=Δi<κCi. Since the diagonal intersection contains the intersection, obviously C is unbounded. Then suppose SC and sup(Sα)=α. Then SCβ for every βα, and since each Cβ is closed, αCβ, so αC.

Title club filter
Canonical name ClubFilter
Date of creation 2013-03-22 12:53:11
Last modified on 2013-03-22 12:53:11
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Definition
Classification msc 03E10
Defines club filter