[itex](X,\rho)[/itex] is a pseudometric space(adsbygoogle = window.adsbygoogle || []).push({});

Define:

x~y if and only if [itex]ρ(x,y)=0[/itex]

(It is shown that x~y is an equivalence relation)

Ques:

If [itex]X^{*}[/itex] is a set of equivalence classes under this relation, then [itex]\rho(x,y)[/itex] depends only on the equivalence classes of x and y and [itex]\rho[/itex] induces a metric on [itex]X^{*}[/itex].

Attempt:

I know that from the question,

[itex]X^{*}=[/itex] {[a]; [itex]a\in X[/itex]} where [itex][a]={x\in X;\rho(x,a)=0}[/itex]

But I don't know how to go about proving that [itex]\rho(x,y)[/itex] depends only on [x] and [y]. I know i need to prove that [itex]\rho(x,y)[/itex] only depends on the all the [itex]c\in X[/itex] such that [itex]\rho(c,x)=\rho(c,y)=0[/itex].

But I just don't know where to start...

Thanks

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equivalence classes and Induced metric

Loading...

Similar Threads for Equivalence classes Induced | Date |
---|---|

Non-Euclidean geometry and the equivalence principle | Jan 31, 2016 |

A question on defining vectors as equivalence classes | Feb 5, 2015 |

Ray vs. Line: Equivalent Freedoms? | Sep 15, 2012 |

Tangent vectors as equivalence classes of curves | Aug 4, 2011 |

Equivalence relations and equivalence classes | Feb 22, 2010 |

**Physics Forums - The Fusion of Science and Community**