Exercise 8.2.7

Question

Describe the $\epsilon$-neighborhoods in $\mathbf{R}^2$ for each of the different metrics described in Exercise 8.2.1. How about for the discrete metric?

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Since this is the Euclidean distance, a $\epsilon$-neighborhood is a circle (not including its boundary) with radius $\epsilon$.
\item An $\epsilon$-neighborhood of $x$ looks like a square with side length $2\epsilon$ centred at $x$.
 \end{enumerate}

For the discrete metric, if $\epsilon > 1$ then an $\epsilon$-neighborhood is the entire plane, while if $\epsilon \leq 1$ a neighborhood of $x$ is just the single point $x$.

Exercise 8.2.7

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Exercise 8.2.7

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