## Definition ## The type of __real numbers__ $\mathbb{R}$ is a [[locally (-1)-connected]] [[Hausdorff]] [[sober]] [[Archimedean ordered field]] with a [[compact]] [[real unit interval]] $[0, 1]$. ## Other types called 'real numbers' ## There are many other different types which are called real numbers in the literature, many of which are do not satisfy the same properties as listed above for the real numbers. These include: * [[Cauchy real numbers (disambiguation)]] * [[Dedekind real numbers (disambiguation)]] * [[Eudoxus real numbers]] * [[localic real numbers]] * [[MacNeille real numbers]] or Dedekind-MacNeille real numbers * [[real unit interval]] based real numbers * [[Euclidean real numbers]] or Escardo-Simpson real numbers * The various types of real numbers defined by Peter Freyd using various definitions of the co-algebraic real unit interval. * [[Stolzenberg real numbers]] from [[closed interval]]s of [[rational numbers]], originally defined by Gabriel Stolzenberg. ## References ## * Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013) * Andrej Bauer and Paul Taylor, [The Dedekind Reals in Abstract Stone Duality](https://www.paultaylor.eu/ASD/dedras/index.html) * Mark Bridger, [Real Analysis: A Constructive Approach Through Interval Arithmetic](https://bookstore.ams.org/amstext-38), Pure and Applied Undergraduate Texts 38, American Mathematical Society, 2019.