Knaster's condition

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. An analogous 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($\omega_1$), ccc implies Knaster's condition, making the two equivalent.

References

• Fremlin, David H. (1984). Consequences of Martin's axiom. Cambridge tracts in mathematics, no. 84. Cambridge: Cambridge University Press. ISBN 0-521-25091-9.