Editing Cardinality of the continuum (section)

==Sets with cardinality of the continuum==

A great many sets studied in mathematics have cardinality equal to <math>{\mathfrak c}</math>. Some common examples are the following:

{{unordered list
|the [[real number]]s <math>\mathbb{R}</math>
|any ([[Degeneracy (mathematics)|nondegenerate]]) closed or open [[Interval (mathematics)|interval]] in <math>\mathbb{R}</math> (such as the [[unit interval]] {{nowrap|<math>[0,1]</math>)}}
|the [[irrational number]]s
|the [[transcendental numbers]]
{{pb}}
The set of real [[algebraic number]]s is countably infinite (assign to each formula its [[Gödel numbering|Gödel number]].) So the cardinality of the real algebraic numbers is {{nowrap|<math>\aleph_0</math>.}} Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is {{nowrap|<math>\mathbb R</math>.}} Thus, since the cardinality of <math>\mathbb R</math> is {{nowrap|<math>\mathfrak c</math>,}} the cardinality of the real transcendental numbers is {{nowrap|<math>\mathfrak c - \aleph_0 = \mathfrak c</math>.}} A similar result follows for complex transcendental numbers, once we have proved that {{nowrap|<math>\left\vert \mathbb{C} \right\vert = \mathfrak c</math>.}}
|the [[Cantor set]]
|[[Euclidean space]] <math>\mathbb{R}^n</math><ref name=Gouvea>[http://www.maa.org/sites/default/files/pdf/pubs/AMM-March11_Cantor.pdf Was Cantor Surprised?], [[Fernando Q. Gouvêa]], ''[[American Mathematical Monthly]]'', March 2011.</ref>
|the [[complex number]]s <math>\mathbb{C}</math>
{{pb}}
Per Cantor's proof of the cardinality of Euclidean space,<ref name=Gouvea /> {{nowrap|<math>\left\vert \mathbb{R}^2 \right\vert = \mathfrak c</math>.}} By definition, any <math>c\in \mathbb{C}</math> can be uniquely expressed as <math>a + bi</math> for some {{nowrap|<math>a,b \in \mathbb{R}</math>.}} We therefore define the bijection
{{block indent|<math>\begin{align}
 f\colon \mathbb{R}^2 &\to \mathbb{C}\\
 (a,b) &\mapsto a+bi
\end{align}</math>}}
|the [[power set]] of the [[natural number]]s <math>\mathcal{P}(\mathbb{N})</math> (the set of all subsets of the natural numbers)
|the set of [[sequences]] of integers (i.e. all functions {{nowrap|<math>\mathbb{N} \rightarrow \mathbb{Z}</math>,}} often denoted {{nowrap|<math>\mathbb{Z}^\mathbb{N}</math>)}}
|the set of sequences of real numbers, {{nowrap|<math>\mathbb{R}^\mathbb{N}</math>}}
|the set of all [[continuous function|continuous]] functions from <math>\mathbb{R}</math> to <math>\mathbb{R}</math>
|the [[Euclidean topology]] on <math>\mathbb{R}^n</math> (i.e. the set of all [[open set]]s in {{nowrap|<math>\mathbb{R}^n</math>)}}
|the [[Borel algebra|Borel σ-algebra]] on <math>\mathbb{R}</math> (i.e. the set of all [[Borel set]]s in {{nowrap|<math>\mathbb{R}</math>).}}
}}