The Discrete Metric

Topology · Axiom Academy

Three worked examples exploring the simplest yet most surprising metric space The discrete metric is the simplest metric you can define on any set. It treats all distinct points as being "unit distance" apart, while identical points are distance zero apart. Despite its simplicity, this metric reveals profound insights about topology and metric spaces. Let be any non-empty set. The discrete metric is defined by: In words: the distance between two points is 0 if they're the same point, and 1 if they're different. To prove is a metric, we must verify three axioms for all : Axiom 1: Positive Definiteness By definition of the discrete metric: The discrete metric only takes values 0 and 1. We consider all cases: If : Then . But the right side is at least 0, so the inequality holds. ✓ If : Then either or , so the right side is at least 1, and the inequality holds. ✓ Therefore, the triangle inequality holds in all cases. ✓ All three axioms are satisfied, so the discrete metric is indeed a valid metric on any set X. 2 Open Balls in the Discrete Metric An open ball of radius centered at is defined as: Let's examine what this means for different values of in the discrete metric. For any , what points satisfy ? Since we need , and for any different point gives , we conclude: The ball contains only the center point! Since the discrete metric only takes values 0 and 1: The ball contains the entire space! Open balls in discrete metric: singleton for r ≤ 1, entire space for r > 1

This is the written version of the interactive lesson above. See the full Topology course.