In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
''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
Polish may refer to:
* Anything from or related to Poland, a country in Europe
* Polish language
* Poles, people from Poland or of Polish descent
* Polish chicken
*Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screenwr ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Bronisław Knaster
Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław.
He is known for his work in point-set topology and in particular for his discoveries in 1922 of ...
.
Knaster's condition implies the
countable chain condition In order theory, a partially ordered set ''X'' is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in ''X'' is countable.
Overview
There are really two conditions: the ''upwards'' and ''downwards'' countable c ...
(ccc), and it is sometimes used in conjunction with a weaker form of
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...
, 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 mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, in which case it means that the
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
(as in, the family of all open sets) with
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabilitie ...
satisfies the condition.
Furthermore, assuming
MA(
), ccc implies Knaster's condition, making the two equivalent.
References
*
Order theory
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