If I'm given a metric, say the discrete metric $\text d_0(x,y):=\begin{cases}
0, & \text{if }\vec x= \vec y\\
1, & \text{if } \vec x \neq \vec y\
\end{cases}$ and want to visualize a set given by $\mathbb{S}^{1}_0(\vec0) = \lbrace\vec x \in \mathbb{R}^2 |\text d_0(\vec x, \vec 0) = 1\rbrace$, how can I grasp the of definition of the set? Since I can't even begin to imagine it, any hint would be highly appreciated.
How to visualize a set in metric space
analysiscalculusdiscrete geometrymetric-spaces
Best Answer
For most metrics you will encounter on ${\mathbb R}^2$ you can sketch sets fairly easily. Start off by working out which points must lie on the boundary of the set you want to sketch and work from there, checking if there are 'holes' in the interior somehow.
For the example you give, since every point except $0$ is a distance $1$ from $0$ the 'sphere' in this case is the whole of ${\mathbb R}^2$. You can sketch that as the usual plane for ${\mathbb R}^2$, with the centre point $0$ highlighted to indicate its exclusion.
You might also want to look at the shapes of the unit ball under various metrics and how they move from diamond-shape to circular to square-shaped.