Can a disk be defined with a discrete metric?

Question

$D=\{(x,y)\in \mathbb{R^2}:\rho_d(x,y)<\epsilon\}$? Say for $\epsilon<1$, the disk doesn't exist.

Edit: Elaborating on the question. Can you call it a disk if it is not the euclidean metric?

Answer 1

An open disc $D_\epsilon(x)$ around $x$ with radius $\epsilon>0$ in e.g. $\mathbb{R}^2$ is defined to be $$D_\epsilon(x):=\{y\in\mathbb{R}^2|d(x,y)<\epsilon\}$$

In case $d$ is the discrete metric this means you have for all $x,y$

$$d(x,x) = 0$$

$$d(x,y) = 1 \text{ if } x\neq y$$

In case $\epsilon < 1$ you have $$D_\epsilon(x) = \{x\}$$ since all $y\neq x$ have $d(x,y)=1$ and are thereby not in $D_\epsilon(x)$.