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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.

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A subset $S$ of the unit sphere has covering radius $r$ if each point of the sphere lies at distance at most $r$ from some point in $S$. There is no known formula which gives the minimum number of points required in a subset with covering radius $r$. Nonetheless it is an important problem, and you will find a table at http://www2.research.att.com/~njas/coverings/index.html (which reports on results of Hardin, Sloane and Smith. Neil Sloane's web [age is a natural place to start if you want more information.

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