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:
Dedekind real numbers (disambiguation page)
localic real numbers? (opens are binary relations, type is a frame and lies in a higher universe in the hierarchy) and sigma-localic real numbers? (opens are functions into Sierpinski space, type is a $\sigma$-frame and lies in the same universe)
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.
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Andrej Bauer and Paul Taylor, The Dedekind Reals in Abstract Stone Duality