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The type of real numbers is a locally (-1)-connected? Hausdorff sober? Archimedean ordered field with a compact? real unit interval? .
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:
localic real numbers?
MacNeille real numbers? or Dedekind-MacNeille real numbers
real unit interval? based real numbers
pre-algebra real numbers, real numbers as usually defined in school mathematics.
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?rational closed intervals , from where the Dedekind real numbers are the located rational closed intervals, and the Cauchy real numbers are the rational closed intervals with a closed locator interval s . ofrational numbers, originally defined by Gabriel Stolzenberg.
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Andrej Bauer and Paul Taylor, The Dedekind Reals in Abstract Stone Duality
Mark Bridger, Real Analysis: A Constructive Approach Through Interval Arithmetic, Pure and Applied Undergraduate Texts 38, American Mathematical Society, 2019.