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.
Is this supposed to be min instead of inf (or am I misunderstanding the notation)?
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.