In article by Boronski, Clark, Oprocha, authors say the following:
Let $D$ be metric on $\mathbb{R}^d\setminus\{0\}$ defined as $D(x, y) = \parallel x - y\parallel + \lvert c(x) - c(y) \rvert$, where $c(y) = \frac{y_1}{\sqrt \sum_{i = 1}^d y_i^2}$. The completion of $\mathbb{R}^d \setminus \{0\}$ with respect to $D$ has remainder given by an interval that can naturally be identified with $[-1, 1]$, corresponding to the limiting value of $c$ for the points in the remainder.
My question is, what they mean by remainder and why it can be identified with goven closed interval.
As far as I know the topological folklore, a remainder of a dense subspace $X$ of a space $Y$ is a space $X\setminus Y$. In this definition $Y$ for me usually a is compactification of a space $X$.
Now put $X=(\mathbb R^d\setminus\{0\},D) $. The remainder $DX=Y\setminus X$ of a standard completion $Y$ of $X$ with respect to $D$ consists of equivalency classes $[x_n]$ of non-convergent in $X$ Cauchy sequences $\{x_n\}$ of points of $X$, where $\{x_n\}\sim \{y_n\}$ iff $\lim_{n\to\infty} D(x_n,y_n)=0$. Since $D(x,y)\ge \|y-z\|$ for each $y,z\in Z$, each Cauchy in $X$ sequence $\{x_n\}$ converges in a space $(\mathbb R^d,\|\cdot\| )$ to some point $x$. If $x\ne 0$ then, by the continuity of the function $c$ in $(\mathbb R^d\setminus\{0\},\|\cdot\| )$, $\{x_n\}$ converges to $x$ also in $X$, so $[x_n]$ is not in the remainder. Thus it remains to consider the case when $x=0$. It is easy to see that sequences $\{x_n\}$ and $\{y_n\}$ of points of $\mathbb R^d\setminus\{0\}$, both converging to $0$ in $(\mathbb R^d\setminus\{0\},\|\cdot\|)$ and Cauchy in $X$, are equivalent iff $\lim_{n\to\infty} c(x_n)= \lim_{n\to\infty} c(y_n)$. So the equivalence classes $[x_n]$ of such sequences are in a bijection $f$ with points of $[-1,1]$, where $f([x_n])= \lim_{n\to\infty} c(y_n)$ for any sequence $\{y_n\}\in [x_n]$. It remains to note that $f$ is also the isometry, that is for any such sequences $\{x_n\}$ and $\{y_n\}$, $\hat D([x_n],[y_n])=|f([x_n])-f([y_n]) |$, where $\hat D$ is the extension of the metric $D$ to $Y$.