Someone I know said "I wish no matter where I am, there is always a place near me so I can visit".
I started to wonder what is the minimum number of places required if he give me what he consider as "near".
I formalized it into a math problem:
Every point on the unit sphere has distance at most $d$ to some point in the set $S$, what is the lower bound for $|S|$?
I also wonder if there are any studies on the generalized version: replace unit sphere with any totally bounded spaces in the problem above.