**metric space**if and only if

(1) | $X$ is a D11: Set |

(2) | $d$ is a D58: Metric on $X$ |

(3) | $\mathcal{T}_d$ is the D444: Metric topology on $X$ with respect to $d$ |

▾ 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

▾ Metric

▾ 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

▾ Metric

Formulation 0

An D5107: Triple $M = (X, \mathcal{T}_d, d)$ is a **metric space** if and only if

(1) | $X$ is a D11: Set |

(2) | $d$ is a D58: Metric on $X$ |

(3) | $\mathcal{T}_d$ is the D444: Metric topology on $X$ with respect to $d$ |

Child definitions

» D65: Cauchy sequence

» D4468: Empty metric space

» D47: Lipschitz map

» D1224: Set diameter

» D1400: Set distance

» D4475: Trivial metric space

» D4468: Empty metric space

» D47: Lipschitz map

» D1224: Set diameter

» D1400: Set distance

» D4475: Trivial metric space

Results