# Difference between revisions of "Pseudo-metric"

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− | ''on a set | + | ''on a set $X$'' |

− | A non-negative real-valued function | + | A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ (that is, on $X \times X$) and satisfying the following three conditions, called the axioms for a pseudo-metric: |

− | a) if | + | a) if $x = y$, then $d(x,y) = 0$; |

− | b) | + | b) $d(x,y) = d(y,x)$ (symmetry); |

− | c) | + | c) $d(x,y) \le d(x,z) + d(z,y)$ (triangle inequality), where $x,y,z$ are arbitrary elements of $X$. |

− | It is not required that | + | It is not required that $d(x,y) = 0$ implies $x=y$. A topology on $X$ is determined by a pseudo-metric $d$ on $X$ as follows: A point $x$ belongs to the closure of a set $A \subseteq X$ if $d(x,A) = 0$, where |

+ | $$ | ||

+ | d(x,A) = \inf_{a \in A} d(x,a) \ . | ||

+ | $$ | ||

− | + | This topology is [[Completely-regular space|completely regular]] but is not necessarily [[Hausdorff space|Hausdorff]]: [[singleton]] sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures. | |

− | |||

− | This topology is completely regular but is not necessarily Hausdorff: singleton sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures. | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Springer (1975)</TD></TR></table> | + | <table> |

− | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Graduate Texts in Mathematics '''27''' Springer (1975) ISBN 0-387-90125-6 {{ZBL|0306.54002}}</TD></TR> | |

− | + | </table> | |

====Comments==== | ====Comments==== | ||

− | See also [[ | + | See also [[Metric]], [[Quasi-metric]] and [[Symmetry on a set]]. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. 532</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. 532 {{ZBL|0141.39401}}</TD></TR> | ||

+ | </table> | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 07:22, 12 December 2016

*on a set $X$*

A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ (that is, on $X \times X$) and satisfying the following three conditions, called the axioms for a pseudo-metric:

a) if $x = y$, then $d(x,y) = 0$;

b) $d(x,y) = d(y,x)$ (symmetry);

c) $d(x,y) \le d(x,z) + d(z,y)$ (triangle inequality), where $x,y,z$ are arbitrary elements of $X$.

It is not required that $d(x,y) = 0$ implies $x=y$. A topology on $X$ is determined by a pseudo-metric $d$ on $X$ as follows: A point $x$ belongs to the closure of a set $A \subseteq X$ if $d(x,A) = 0$, where $$ d(x,A) = \inf_{a \in A} d(x,a) \ . $$

This topology is completely regular but is not necessarily Hausdorff: singleton sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures.

#### References

[1] | J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002 |

#### Comments

See also Metric, Quasi-metric and Symmetry on a set.

#### References

[a1] | E. Čech, "Topological spaces" , Interscience (1966) pp. 532 Zbl 0141.39401 |

**How to Cite This Entry:**

Pseudo-metric.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=29453