Primitives

This note covers basic constructs in geometry.

Metric Space

definition. Let M{M} be a set and let d{d} be a function of the form d:(M×M)↦R.{d:(M \times M) \mapsto \reals.} A metric space is an ordered pair (M,d){(M,d)} that satisfies the following axioms, for all x,y,z∈M:{x,y,z \in M:}

  1. d(x,x)=0.{d(x,x)=0.}
  2. If x≠y,{x \neq y,} then d(x,y)>0.{d(x,y) \gt 0.}
  3. d(x,y)=d(y,x).{d(x,y)=d(y,x).}
  4. d(x,z)≤d(x,y)+d(y,z).{d(x,z) \le d(x,y)+d(y,z).}

Points

definition. Given a metric space (M,d){(M,d)} we call each element of M×M{M \times M} a point.