It’s supposed to be inf (the infimum). Which is the same as the minimum whenever the minimum exists, but sometimes it doesn’t exist.
Suppose S is (0,1), i.e.{x∈R:0<x<1} and the point p is 3. Then the set {d(p,q)|q∈S} doesn’t have a smallest element. Something like d(0.9999,3) is pretty close but you can always find a pair that’s even closer. So the distance is defined as the largest lower-bound on the set {d(p,q)|q∈S}, which is the infimum, in this case 2.
It’s supposed to be inf (the infimum). Which is the same as the minimum whenever the minimum exists, but sometimes it doesn’t exist.
Suppose S is (0,1), i.e.{x∈R:0<x<1} and the point p is 3. Then the set {d(p,q)|q∈S} doesn’t have a smallest element. Something like d(0.9999,3) is pretty close but you can always find a pair that’s even closer. So the distance is defined as the largest lower-bound on the set {d(p,q)|q∈S}, which is the infimum, in this case 2.