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This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to certain chains of open sets in topological spaces and chains in complete
Boolean algebras, where chain conditions sometimes happen to be equivalent to
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antichain conditions. For example, if κ is a cardinal, then in a complete
Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions.
81:, ccc partial orders are used because forcing with any generic set over such an order preserves cardinals and cofinalities. Furthermore, the ccc property is preserved by finite support iterations (see
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countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound.
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Products of
Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp. 398–403 (1964)
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Partial orders and spaces satisfying the ccc are used in the statement of
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In general, a ccc topological space need not be separable. For example,
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More generally, if κ is a cardinal then a poset is said to satisfy the
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Forcing (set theory) § The countable chain condition
308:, Springer Monographs in Mathematics, Berlin, New York:
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An example of a ccc topological space is the real line.
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separable spaces is a separable space and, thus, ccc.
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is said to satisfy the countable chain condition, or
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193:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}
264:{\displaystyle \{0,1\}^{2^{2^{\aleph _{0}}}}}
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55:There are really two conditions: the
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104:Examples and properties in topology
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306:Set Theory: Millennium Edition
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282:Paracompact ccc spaces are
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319:978-3-540-44085-7
133:pairwise disjoint
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77:In the theory of
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328:Kunen, Kenneth.
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340:Categories
294:References
279:separable.
271:with the
247:ℵ
180:ℵ
117:Condition
61:downwards
45:countable
304:(2003),
284:Lindelöf
115:Suslin's
51:Overview
79:forcing
57:upwards
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147:Every
131:every
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129:i.e.
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