Metric Spaces
Topological Worlds with Distance Functions
A metric is a function \(d(x,y) \) that calculates the distance between two points in a space. This function returns a real number and must satisfy a few axioms:
\(d(x,y)>0\). The distance between two different points must be positive.
\(d(x,x)=0\). The distance between a point and itself is 0. I shudder to imagine a world where this were not true.
\(d(x,z)\leq d(x,y) + d(y,z)\): This is called the triangle inequality and feels different than the first two axioms, so let's discuss it in greater detail below.
\( d(x,z) \leq d(x,y) + d(y,z) \)
A Circle