Euclid defined the real numbers with geometry.
The definition of real numbers in terms of Cauchy sequences is algebraic. The definition in terms of Dedekind cuts is also algebraic but it has more of a geometric flavor. The best definition of R is
This is basically the topological definition of R, and it is pretty much geometric. Topology is the study of things by examination of their open subsets, and the best definition of R is one big open set. So, the topology approach is the best approach to R.
If you ask an algebraist what a 4D shape is, they might tell you it's a tesseract. Is the 4D universe a tesseract like the black hole special effects sequence in Interstellar? No! That's a 4D shape in the Euclidean topology while the universe is 4D in the Lorentzian topology. The 4D Lorentzian shape is a hypercone, which is like an ice cream cone. The hypercone has a scoop of ice cream at each height of the ice cream cone, and you can see how that's like the expanding universe: the scoop of ice cream has to get bigger as the opening on the cone gets wider. In 3D, if you did that with a real ice cream cone, the scoops would overlap, and you'd need an infinite number of scoops anyhow because there is a continuous spectrum of cone heights, each one with a scoop. In 4D, the scoops don't overlap. There is an infinite number of 3D scoops in a hypercone like like there is an infinite number of 2D squares in a cube. So topology makes it very easy to understand the 4D shape of the universe because it is easier to understand an ice cream cone than to understand a tesseract. Topology is a meme in a good way. Very often the best way to explain something is with informal topology. It is so powerful, pic related little critter got invented to obfuscate it.
