Supremum Metric on Continuous Real Functions/Examples/Closure of Open 1-Ball of 0 on Unit Interval

Example of Supremum Metric on Continuous Real Functions
Let $\closedint 0 1 \subseteq \R$ be the closed unit interval.

Let $\mathscr C \closedint 0 1$ be the supremum space of continuous functions $f: \closedint 0 1 \to \R$.

Then:
 * $\map \cl {\map {B_1} \bszero} = \set {f \in \mathscr C \closedint 0 1: \map {d_\infty} {f, \bszero} \le 1}$

where:
 * $\map {B_1} \bszero$ denotes the open $1$-ball of $\bszero$
 * $\d_\infty$ denotes the Chebyshev distance
 * $\bszero$ denotes the constant function $f_0$.

Proof
Let $S = \set {f \in \mathscr C \closedint 0 1: \map {d_\infty} {f, \bszero} \le 1}$.

We have that:
 * $\map \cl {\map {B_1} \bszero} \subseteq S$

Suppose $f \in S$.

Let $\epsilon \in \R_{>0}$ be given.

Then:
 * $\paren {1 - \dfrac \epsilon 2} \in \map {B_1} \bszero \cap \map {b_\epsilon} f$

and so:
 * $f \in \map \cl {\map {B_1} \bszero}$