In mathematics, a collapsing algebra is a type of

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...

sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by

Azriel Lévy Azriel Lévy (Hebrew: עזריאל לוי; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem. Biography Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, und ...

in 1963. The collapsing algebra of λ^ω is a

complete Boolean algebra In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolea ...

with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no

free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...

complete Boolean algebra on a countable number of elements.

Definition

There are several slightly different sorts of collapsing algebras. If κ and λ are cardinals, then the Boolean algebra of

regular open set A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if \operatorname(\overline) = S or, equivalently, if \partial(\overline)=\partial S, where \operatorname S, \ov ...

s of the product space κ^λ is a collapsing algebra. Here κ and λ are both given the discrete topology. There are several different options for the topology of κ^λ. The simplest option is to take the usual product topology. Another option is to take the topology generated by open sets consisting of functions whose value is specified on less than λ elements of λ.

References

* * * {{algebra-stub Boolean algebra Forcing (mathematics)