In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

 P is said to have Knaster's condition upwards (sometimes property (K)) if any uncountable subset A of P has an upwards-linked

Linked set

In mathematics, an upwards linked set A is a subset of a partially ordered set P, in which any two of elements A have a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound. Note that every centered set is linked, which includes, in particular,...

 uncountable subset. Anologous definition applies to Knaster's condition downwards.

The property is named after Polish

Poles

thumb|right|180px|The state flag of [[Poland]] as used by Polish government and diplomatic authoritiesThe Polish people, or Poles , are a nation indigenous to Poland. They are united by the Polish language, which belongs to the historical Lechitic subgroup of West Slavic languages of Central Europe...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Bronisław Knaster.

Knaster's condition implies ccc

CCC

- Businesses and organizations :* Canterbury of New Zealand, a New Zealand-based sports apparel company* Center for Community Change, one of the larger community building organizations in the United States* Color Climax Corporation, a Danish pornography company...

, and it is sometimes used in conjunction with a weaker form of Martin's axiom

Martin's axiom

In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...

, 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

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, in which case it means that the topology (as in, the family of all open sets) with inclusion

Inclusion

Inclusion may refer to:- Metallurgy :*Inclusion , a type of metal casting defect*Inclusions in Aluminium Alloys, solid particles in liquid aluminium alloy- Social inclusion of persons :...

 satisfies the condition.

Furthermore, assuming MA

Martin's axiom

In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...

(), ccc implies Knaster's condition, making the two equivalent.