ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Basic binary operation
Unsigned basic binary operation
Semimetric
Definition D58
Metric
Formulation 0
Let $X$ be a D11: Set.
An D5134: Unsigned basic binary operation $d : X \times X \to [0, \infty]$ is a metric on $X$ if and only if
(1) $\forall \, x, y, z \in X : d(x, z) \leq d(x, y) + d(y, z)$
(2) $\forall \, x, y \in X : d(x, y) = d(y, x)$ (D3490: Symmetric function)
(3) \begin{equation} \forall \, x, y \in X \left( d(x, y) = 0 \quad \iff \quad x = y \right) \end{equation}
Children
D1107: Metric space
D4473: Trivial metric
Results
R102: Singletons are closed in a metric space