Let $(X,d)$ be a metric space and let $A\subset X$. If $A$ is either open or closed, then $(\partial A)^{\circ} = \varnothing$. I am asked to find a metric space and a subset that the interior of its boundary is not empty. I have tried something with the discrete metric, but then realized that since every set in a discrete metric is a union of open sets, every set is open, so this has no future.
[Math] A set which the interior of its boundary is not empty
general-topologymetric-spaces
Best Answer
You're right that a discrete metric won't work here. HINT: you're looking for a set with "big" boundary but "small" interior. Work in $\mathbb{R}$ with the usual metric; can you think of a set that "fills out" $\mathbb{R}$, but with lots of "holes"?