How can we show that the intersection of a series of closed unbounded sets is closed unbounded? [duplicate]












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This question already has an answer here:




  • Club filter of $kappa$ is $kappa$-complete

    1 answer




I am currently stuck on an exercise in set-theory and would like to get some help:
If $S$ is a stationary subset of a regular uncountable cardinal $κ$, and a subset $C$ of $κ$ is an S-cub if it is unbounded in $κ$ and $sup(x) ∈ C$
holds for every $x ⊆ C$ with $sup(x) ∈ S$, then how can we prove that for every sequence $(C_α | α < λ)$ of
S-cubs with λ < κ, $bigcap_{α<λ} C_α$ is an S-cub?



Thanks in advance.










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marked as duplicate by amWhy, Holo, Alessandro Codenotti, Cesareo, José Carlos Santos Jan 3 at 1:17


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














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    $begingroup$
    It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
    $endgroup$
    – JDH
    Jan 2 at 22:44










  • $begingroup$
    Why was it closed as a duplicate? It isn't a duplicate.
    $endgroup$
    – JDH
    Jan 3 at 3:14
















1












$begingroup$



This question already has an answer here:




  • Club filter of $kappa$ is $kappa$-complete

    1 answer




I am currently stuck on an exercise in set-theory and would like to get some help:
If $S$ is a stationary subset of a regular uncountable cardinal $κ$, and a subset $C$ of $κ$ is an S-cub if it is unbounded in $κ$ and $sup(x) ∈ C$
holds for every $x ⊆ C$ with $sup(x) ∈ S$, then how can we prove that for every sequence $(C_α | α < λ)$ of
S-cubs with λ < κ, $bigcap_{α<λ} C_α$ is an S-cub?



Thanks in advance.










share|cite|improve this question











$endgroup$



marked as duplicate by amWhy, Holo, Alessandro Codenotti, Cesareo, José Carlos Santos Jan 3 at 1:17


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
    $endgroup$
    – JDH
    Jan 2 at 22:44










  • $begingroup$
    Why was it closed as a duplicate? It isn't a duplicate.
    $endgroup$
    – JDH
    Jan 3 at 3:14














1












1








1


1



$begingroup$



This question already has an answer here:




  • Club filter of $kappa$ is $kappa$-complete

    1 answer




I am currently stuck on an exercise in set-theory and would like to get some help:
If $S$ is a stationary subset of a regular uncountable cardinal $κ$, and a subset $C$ of $κ$ is an S-cub if it is unbounded in $κ$ and $sup(x) ∈ C$
holds for every $x ⊆ C$ with $sup(x) ∈ S$, then how can we prove that for every sequence $(C_α | α < λ)$ of
S-cubs with λ < κ, $bigcap_{α<λ} C_α$ is an S-cub?



Thanks in advance.










share|cite|improve this question











$endgroup$





This question already has an answer here:




  • Club filter of $kappa$ is $kappa$-complete

    1 answer




I am currently stuck on an exercise in set-theory and would like to get some help:
If $S$ is a stationary subset of a regular uncountable cardinal $κ$, and a subset $C$ of $κ$ is an S-cub if it is unbounded in $κ$ and $sup(x) ∈ C$
holds for every $x ⊆ C$ with $sup(x) ∈ S$, then how can we prove that for every sequence $(C_α | α < λ)$ of
S-cubs with λ < κ, $bigcap_{α<λ} C_α$ is an S-cub?



Thanks in advance.





This question already has an answer here:




  • Club filter of $kappa$ is $kappa$-complete

    1 answer








set-theory






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edited Jan 2 at 22:23









Davide Giraudo

127k16152266




127k16152266










asked Jan 2 at 21:50









P. DeckerP. Decker

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marked as duplicate by amWhy, Holo, Alessandro Codenotti, Cesareo, José Carlos Santos Jan 3 at 1:17


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by amWhy, Holo, Alessandro Codenotti, Cesareo, José Carlos Santos Jan 3 at 1:17


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 1




    $begingroup$
    It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
    $endgroup$
    – JDH
    Jan 2 at 22:44










  • $begingroup$
    Why was it closed as a duplicate? It isn't a duplicate.
    $endgroup$
    – JDH
    Jan 3 at 3:14














  • 1




    $begingroup$
    It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
    $endgroup$
    – JDH
    Jan 2 at 22:44










  • $begingroup$
    Why was it closed as a duplicate? It isn't a duplicate.
    $endgroup$
    – JDH
    Jan 3 at 3:14








1




1




$begingroup$
It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
$endgroup$
– JDH
Jan 2 at 22:44




$begingroup$
It's not a duplicate of that question, since this question is about the $S$-clubs. The $C_alpha$ may not be club at all, but only $S$-club.
$endgroup$
– JDH
Jan 2 at 22:44












$begingroup$
Why was it closed as a duplicate? It isn't a duplicate.
$endgroup$
– JDH
Jan 3 at 3:14




$begingroup$
Why was it closed as a duplicate? It isn't a duplicate.
$endgroup$
– JDH
Jan 3 at 3:14










1 Answer
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It is clear that we satisfy the closure part, since if a bounded set $x$ with $sup(x)in S$ is contained in the intersection $bigcap_alpha C_alpha$, then $x$ is contained in each $C_alpha$, and so $sup(x)$ is in each $C_alpha$ and hence in the intersection $bigcap_alpha C_alpha$, as desired.



So the only difficult part is to show that $bigcap_alpha C_alpha$ is unbounded in $kappa$. For this, let $bar C_alpha$ be the closure of $C_alpha$, that is, $C_alpha$ with all of its limit points. So this is a club set in $kappa$, and since the intersection of fewer than $kappa$ many clubs is club, it follows that $bigcap_alphabar C_alpha$ is closed and unbounded in $kappa$. Thus, since $S$ is stationary, there are unboundedly many $betain S$ that are limits of $bigcap_alpha bar C_alpha$, and these $beta$'s must all be in every $C_alpha$ since each $C_alpha$ is $S$-closed.






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    1 Answer
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    1 Answer
    1






    active

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    active

    oldest

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    active

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    3












    $begingroup$

    It is clear that we satisfy the closure part, since if a bounded set $x$ with $sup(x)in S$ is contained in the intersection $bigcap_alpha C_alpha$, then $x$ is contained in each $C_alpha$, and so $sup(x)$ is in each $C_alpha$ and hence in the intersection $bigcap_alpha C_alpha$, as desired.



    So the only difficult part is to show that $bigcap_alpha C_alpha$ is unbounded in $kappa$. For this, let $bar C_alpha$ be the closure of $C_alpha$, that is, $C_alpha$ with all of its limit points. So this is a club set in $kappa$, and since the intersection of fewer than $kappa$ many clubs is club, it follows that $bigcap_alphabar C_alpha$ is closed and unbounded in $kappa$. Thus, since $S$ is stationary, there are unboundedly many $betain S$ that are limits of $bigcap_alpha bar C_alpha$, and these $beta$'s must all be in every $C_alpha$ since each $C_alpha$ is $S$-closed.






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      It is clear that we satisfy the closure part, since if a bounded set $x$ with $sup(x)in S$ is contained in the intersection $bigcap_alpha C_alpha$, then $x$ is contained in each $C_alpha$, and so $sup(x)$ is in each $C_alpha$ and hence in the intersection $bigcap_alpha C_alpha$, as desired.



      So the only difficult part is to show that $bigcap_alpha C_alpha$ is unbounded in $kappa$. For this, let $bar C_alpha$ be the closure of $C_alpha$, that is, $C_alpha$ with all of its limit points. So this is a club set in $kappa$, and since the intersection of fewer than $kappa$ many clubs is club, it follows that $bigcap_alphabar C_alpha$ is closed and unbounded in $kappa$. Thus, since $S$ is stationary, there are unboundedly many $betain S$ that are limits of $bigcap_alpha bar C_alpha$, and these $beta$'s must all be in every $C_alpha$ since each $C_alpha$ is $S$-closed.






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        It is clear that we satisfy the closure part, since if a bounded set $x$ with $sup(x)in S$ is contained in the intersection $bigcap_alpha C_alpha$, then $x$ is contained in each $C_alpha$, and so $sup(x)$ is in each $C_alpha$ and hence in the intersection $bigcap_alpha C_alpha$, as desired.



        So the only difficult part is to show that $bigcap_alpha C_alpha$ is unbounded in $kappa$. For this, let $bar C_alpha$ be the closure of $C_alpha$, that is, $C_alpha$ with all of its limit points. So this is a club set in $kappa$, and since the intersection of fewer than $kappa$ many clubs is club, it follows that $bigcap_alphabar C_alpha$ is closed and unbounded in $kappa$. Thus, since $S$ is stationary, there are unboundedly many $betain S$ that are limits of $bigcap_alpha bar C_alpha$, and these $beta$'s must all be in every $C_alpha$ since each $C_alpha$ is $S$-closed.






        share|cite|improve this answer











        $endgroup$



        It is clear that we satisfy the closure part, since if a bounded set $x$ with $sup(x)in S$ is contained in the intersection $bigcap_alpha C_alpha$, then $x$ is contained in each $C_alpha$, and so $sup(x)$ is in each $C_alpha$ and hence in the intersection $bigcap_alpha C_alpha$, as desired.



        So the only difficult part is to show that $bigcap_alpha C_alpha$ is unbounded in $kappa$. For this, let $bar C_alpha$ be the closure of $C_alpha$, that is, $C_alpha$ with all of its limit points. So this is a club set in $kappa$, and since the intersection of fewer than $kappa$ many clubs is club, it follows that $bigcap_alphabar C_alpha$ is closed and unbounded in $kappa$. Thus, since $S$ is stationary, there are unboundedly many $betain S$ that are limits of $bigcap_alpha bar C_alpha$, and these $beta$'s must all be in every $C_alpha$ since each $C_alpha$ is $S$-closed.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 2 at 22:11

























        answered Jan 2 at 22:05









        JDHJDH

        32.6k680145




        32.6k680145















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