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$\kappa$-complete (Definition)

A structured set $ S$ (typically a filter or a Boolean algebra) is $ \kappa$-complete if, given any $ K\subseteq S$ with $ \vert K\vert<\kappa$, $ \bigcap K\in S$. It is complete if it is $ \kappa$-complete for all $ \kappa$.

Similarly, a partial order is $ \kappa$-complete if any sequence of fewer than $ \kappa$ elements has an upper bound within the partial order.

A $ \aleph_1$-complete structure is called countably complete.



"$\kappa$-complete" is owned by Henry.
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See Also: filter, Boolean algebra

Other names:  kappa-complete, kappa complete
Also defines:  countably complete
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Cross-references: upper bound, sequence, partial order, complete, Boolean algebra, filter

This is version 8 of $\kappa$-complete, born on 2002-07-29, modified 2003-01-31.
Object id is 3230, canonical name is KappaComplete2.
Accessed 4895 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
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