Proposition 1

Every open (resp. closed) ball of $\mathbf{R}^n$ is an open (resp. compact) set. The closure of an open ball is the closed ball with the same centre and the same radius; the interior of a closed ball is the open ball with the same centre and the same radius.

The open (resp. closed) ball with centre $x_0$ and radius $r$ is the inverse image of the interval ]$-\infty, r[$ (resp. ]$-\infty, r]$) under the continuous function $d(x_0, x)$; it is therefore open (resp. closed and bounded, hence compact). If $d(x_0, x) = r$, and if $y = x_0 + t(x - x_0)$ ($0 < t < 1$) is a point of the open segment with end-points $x_0$ and $x$, we have $d(x_0, y) = tr < r$, and $d(x, y) = (1 - t)r$ is as small as we please; hence $x$ lies in the closure of the open ball with centre $x_0$ and radius $r$. Again, if $z = x + t(x - x_0)$ ($t > 0$) is a point of the open ray with origin $x$ and direction vector $x - x_0$, we have

and $d(x, z) = tr$ is as small as we please; hence $x$ is not an interior point of the closed ball with centre $x_0$ and radius $r$.