Uncountable set information

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. Although the terms...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. While...

a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of...

that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of ω...

considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set.[further explanation needed] Suppose A {\displaystyle...

{\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers)...

Cantor set a universal probability space in some ways. In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has...

\end{cases}}} The set of all such indicator functions, { 1 r } r ∈ R {\displaystyle \{\mathbf {1} _{r}\}_{r\in \mathbb {R} }} , is an uncountable set indexed by...

infinite set of components is covered formally by allowing n = ∞ {\displaystyle n=\infty \!} . Where the set of component distributions is uncountable, the...

existence theorem that there are such sets. Each Vitali set is uncountable, and there are uncountably many Vitali sets. The proof of their existence depends...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

there is an uncountable set of indiscernibles for some Lα, and the phrase "0# exists" is used as a shorthand way of saying this. A closed set I {\displaystyle...

sets, αB will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal...

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set. Another set F with the...

strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero...

cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally...

terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction...

the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous...

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple...

algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations...

countable ones, so it is an uncountable set. Therefore, ℵ1 is distinct from ℵ0. The definition of ℵ1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom...