Metric: Difference between revisions
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== Definitions == | == Definitions == | ||
Let <math>X</math> be a given non-empty set. A | Let <math>X</math> be a given non-empty set. A metric on <math>X</math> is a function <math display="block">d: X\times X\to \mathbb{R},</math> | ||
that satisfies the following four properties for all <math>x,y,z\in X</math>: | that satisfies the following four properties for all <math>x,y,z\in X</math>: | ||
Revision as of 20:03, 12 June 2026
In mathematics, a metric or distance function is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric space.
Definitions
Let 
be a given non-empty set. A metric on is a function
that satisfies the following four properties for all 
:
- Non-negativity:
- Identity of indiscernibles:
- Symmetry:
- Triangle inequality:
.
Properties
Property.
Non-negativity is implied by the other three properties.
Proof
By the identity of indisscernbles, 
. Thus
By symmetry, . Thus
Let 
. By triangle inequality,
. Thus
. Thus
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