Exercise 8.2.12

Question

\begin{enumerate}[label=(\alph*)] 
\item Show
$$\overline{V_\epsilon(x)} \subseteq \{y \in X : d(x,y) \leq \epsilon\},$$
in an arbitrary metric space $(X, d)$.
\item To keep things from sounding too familiar, find an example of a specific metric space where
$$\overline{V_\epsilon(x)} \neq \{y \in X : d(x,y) \leq \epsilon\}.$$
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Let $y$ be a limit point of $V_\epsilon(x)$. For all $a > 0$, we have some $z \in V_\epsilon(x)$ where $d(z, y) < a$, and thus
$$d(x, y) \leq d(x, z) + d(z, y) < \epsilon + a$$
which implies $d(x, y) \leq \epsilon $ and hence
$$\overline{V_\epsilon(x)} \subseteq \{y \in X : d(x,y) \leq \epsilon\}$$

\item Consider the metric space $(\mathbf{R}, \rho)$ where $\rho$ is the discrete metric, and consider $V_1(0) = \{0\}$. $\overline{V_1(0)} = \{0\}$ but $\{y \in \mathbf{R} : \rho(x,y) \leq 1\} = \mathbf{R}$.
 \end{enumerate}

Exercise 8.2.12

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Exercise 8.2.12

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