Math 2.0

definition

Metric space

Definition

Let XX be a set. A metric on XX is a function d ⁣:X×X→Rd \colon X \times X \to \mathbb{R} with these properties, for all x,y,z∈Xx, y, z \in X:

  1. d(x,x)=0d(x, x) = 0.
  2. If x≠yx \neq y, then d(x,y)>0d(x, y) > 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).

A metric space is a pair (X,d)(X, d), where XX is a set and dd is a metric on XX. The number d(x,y)d(x, y) is the distance between xx and yy.

Remarks

Property 4 is the triangle inequality for a metric. The properties give d(x,y)≥0d(x, y) \ge 0 for all x,yx, y: take z=xz = x in property 4 and use property 3. The set R\mathbb{R} with d(x,y)=∣x−y∣d(x, y) = |x - y| is a metric space: this is the usual metric on the real line. If you remove property 2, you get a pseudometric space.

Explanations

L1 human

A metric is a rule that gives a distance to each pair of points. The four properties are the facts that you expect from distances on a map. For example, the road distance between towns is not always a metric, because one-way roads can break symmetry.

What this node needs for W2
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

144 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Metric_space, Definition

What links here

Used by