A ring is closed under intersection. And delta-ring is closed under countable intersection. According to my understanding, if a family of sets is closed under intersection, than it should also be closed under countable intersection.Let's say R is a ring. To R be a delta-ring, A n ∩ A n + 1 ∩ A n + 2 ∩ . . . should also be in R. And, as a ring is closed under intersection, A n ∩ A n + 1 is in R, then A n ∩ A n + 1 ∩ A n + 2 is in R, and so on. So, any ring is a delta-ring.What am I missing? I guess I don't properly understand what countable intersection is. Thank you for reading!

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A ring is closed under intersection. And delta-ring is closed under countable intersection. According to my understanding, if a family of sets is closed under intersection, than it should also be closed under countable intersection.Let&#039;s say   R is a ring. To   R be a delta-ring,       A    n    ∩      A          n      +      1        ∩      A          n      +      2        ∩  .  .  . should also be in   R. And, as a ring is closed under intersection,       A    n    ∩      A          n      +      1       is in   R, then       A    n    ∩      A          n      +      1        ∩      A          n      +      2       is in   R, and so on. So, any ring is a delta-ring.What am I missing? I guess I don&#039;t properly understand what countable intersection is. Thank you for reading!

Ashley Parks · Accepted Answer

Closure under intersection (i.e. of two elements) implies closure under finite intersection by induction. It does not imply closure under countable (i.e. a type of infinite) intersection.Take a simple example from topology. The intervals            (        −      1    i    ,      1    i              )      for i=1,2,… are open, and it&#039;s known in topology (and reasonably intuitive, I think) that the intersection of finitely many, i.e.      ⋂          i      =      1        n              (        −      1    i    ,      1    i              )      is still an open set. But I think it&#039;s also quite clear that the countable intersection      ⋂          i      =      1              ∞                  (        −      1    i    ,      1    i              )      is not an open set.Note: I should mention that &quot;countable&quot; is sometimes used to mean &quot;countably infinite&quot;, but in certain places it&#039;s used as an umbrella term for &quot;finite or denumerable&quot;, where denumerable means &quot;can be put into bijection with the natural numbers&quot;.

A ring is closed under intersection. And delta-ring is closed under countable intersection. Accordin

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