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 closed intervals, where the Dedekind real numbers are the located Dedekind real closed intervals, and the computable real numbers are the Dedekind real closed intervals with a locator.
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.