Closure (Metric Space)/Examples/Union of Disjoint Closed Real Intervals

Examples of Closure in the context of a Metric Space
Let $\R$ be the real number line under the usual (Euclidean) metric.

Let $M$ be the subspace of $\R$ defined as:
 * $M = \closedint 0 1 \cup \closedint 2 3$

Let $\map {B_1} 1$ denote the open $1$-ball of $1$ in $M$.

Let $\map { {B_1}^-} 1$ denote the closed $1$-ball of $1$ in $M$.

Then:
 * $\map \cl {\map {B_1} 1} = \closedint 0 1$

while:
 * $\map { {B_1}^-} 1 = \closedint 0 1 \cup \set 2$