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| | In [[mathematics]], a [[partially ordered set]] ''P'' is said to have '''Knaster's condition upwards''' (sometimes '''property (K)''') if any [[uncountable]] subset ''A'' of ''P'' has an [[linked set|upwards-linked]] uncountable subset. Anologous definition applies to '''Knaster's condition downwards'''. |
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| | The property is named after [[Poles|Polish]] [[mathematician]] [[Bronisław Knaster]]. |
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| | Knaster's condition implies a [[countable chain condition]] (ccc), and it is sometimes used in conjunction with a weaker form of [[Martin's axiom]], where the ccc requirement is replaced with Knaster's condition. Not unlike ccc, Knaster's condition is also sometimes used as a property of a [[topological space]], in which case it means that the [[topology]] (as in, the family of all open sets) with [[inclusion (set theory)|inclusion]] satisfies the condition. |
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| | Furthermore, assuming [[Martin's axiom|MA]](<math>\omega_1</math>), ccc implies Knaster's condition, making the two equivalent. |
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| | == References == |
| | *{{cite book | last=Fremlin | first=David H. | title=Consequences of Martin's axiom | publisher=[[Cambridge University Press]]| location=Cambridge | year=1984 | isbn=0-521-25091-9 | others=Cambridge tracts in mathematics, no. 84}} |
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| | [[Category:Order theory]] |
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| | {{Mathlogic-stub}} |
Latest revision as of 18:41, 30 January 2014
In mathematics, a partially ordered set P is said to have Knaster's condition upwards (sometimes property (K)) if any uncountable subset A of P has an upwards-linked uncountable subset. Anologous definition applies to Knaster's condition downwards.
The property is named after Polish mathematician Bronisław Knaster.
Knaster's condition implies a countable chain condition (ccc), and it is sometimes used in conjunction with a weaker form of Martin's axiom, where the ccc requirement is replaced with Knaster's condition. Not unlike ccc, Knaster's condition is also sometimes used as a property of a topological space, in which case it means that the topology (as in, the family of all open sets) with inclusion satisfies the condition.
Furthermore, assuming MA(), ccc implies Knaster's condition, making the two equivalent.
References
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