The circle, rotations and the doubling map

The circle is $\mathbb{T}=[0,1)$ with distance $d(x,y)=\min (\mathrm{\mid }x-y\mathrm{\mid },1-\mathrm{\mid }x-y\mathrm{\mid })={\min }_{k\in \mathbb{Z}}\mathrm{\mid }x-y-k\mathrm{\mid }$. The minimum is attained, since $-1<x-y<1$. It is nonnegative and symmetric, and it vanishes precisely when $x=y$. For minimizing integers k,l, $d(x,z)\le \mathrm{\mid }x-z-(k+l)\mathrm{\mid }\le \mathrm{\mid }x-y-k\mathrm{\mid }+\mathrm{\mid }y-z-l\mathrm{\mid }=d(x,y)+d(y,z)$, proving the triangle inequality.

Write $\{u\}=u-\lfloor u\rfloor \in [0,1)$ for fractional part. The rotation of angle $\alpha \in \mathbb{R}$ and the doubling map are

$R_{\alpha }(x)=\{x+\alpha \},D(x)=\{2x\}\mathrm{.}$

They satisfy $d(R_{\alpha }x,R_{\alpha }y)=d(x,y)$ and $d(Dx,Dy)\le 2d(x,y)$ by the integer-minimum formula, so are continuous in the circle metric. This topology differs from the ordinary interval topology at zero: points tending to 1 from below tend to 0 on the circle.

Open circle balls of radius less than $1\mathrm{/}2$ are single ordinary intervals or two intervals meeting the cut at 0 and 1. By Both $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ are dense in $\mathbb{R}$, and every nonempty open subset of $\mathbb{R}$ is uncountable, $\mathbb{Q}$ is countably infinite and A product of two at most countable sets is at most countable, balls with rational centers and positive rational radii form a countable base: given a ball about x, take a rational center close enough to x and a smaller rational radius whose ball contains x and stays inside the original ball. Thus circle-open sets are countable unions of ordinary Borel sets. Conversely ordinary interval-open subsets of $(0,1)$ are circle-open, and the singleton $\{0\}$ is circle-closed. Every relatively open subset of $[0,1)$ is therefore circle-Borel. The two Borel sigma-algebras agree.

For measures we assume countable choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $\lambda$ be the restriction of Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$ to $[0,1)$, either on Borel sets or on Lebesgue measurable sets. The measure and volume clauses of Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume give its total mass one: singletons have measure zero by covering them with intervals of arbitrarily small length, so changing an interval endpoint does not change length. By $\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets, the latter version is the completion of the former (intersect its Borel representatives and null covers with $[0,1)$). Countable choice is used for these measure constructions only; the circle metric and maps above are choice-free.