Talk:Pseudometric space

question about the induced metric

In order for the metric on the equivalence classes to be well defined, doesn't it need to be an infimum over the metric distances between all elements of each equivalence class? Or am I being overly careful?--217.84.30.228 (talk) 19:57, 18 April 2012 (UTC)

Let $x,x'\in [x]$ and $y,y'\in [y]$. Consider $d(x,y)$. Using just the pseudometric properties of $d$ and the fact that $d(x,x')=0$ and $d(y,y')=0$ (by definition), we have $d(x,y)\leq d(x,y')+d(y',y)=d(x,y')=d(y',x)\leq d(y',x')+d(x',x)=d(y',x')=d(x',y')$ so $d(x,y)\leq d(x',y')$. Repeat the same steps but start with $x'$ and $y'$. $d(x',y')\leq d(x',y)+d(y,y')=d(x',y)=d(y,x')\leq d(y,x)+d(x,x')=d(y,x)=d(x,y)$ so $d(x',y')\leq d(x,y)$. Those two results imply $d(x',y')=d(x,y)$ so the definition $d^{*}([x],[y])=d(x,y)$ is well-defined. For what it's worth, it wasn't intuitively obvious to me either. Cheers,Jason Quinn (talk) 06:42, 10 September 2012 (UTC)

Does the completion of metric spaces really fit here? In fact, the construction of complete metric spaces from metric spaces is something different to the construction of metric spaces from pseudometric spaces. Certainly, both cases are based in the definition of an equivalent relation and the related quotient space --ErnstGR (talk) 16:31, 16 October 2014 (UTC)

Useless examples

They are just assertions and not examples where people can see why it is a pseudometric space. — Preceding unsigned comment added by 217.225.42.190 (talk) 17:59, 30 October 2012 (UTC)

Is Minkowski space-time (in physics) a pseudometric space? That would seem like an important example to use. Cesiumfrog (talk) 06:12, 9 October 2014 (UTC)

No, it is not. The definition of pseudometric implies $d(x,y)\not <0$--ErnstGR (talk) 14:50, 16 October 2014 (UTC)

Yes it is.

Take a pseudometric space with signature (3,1), like spacetime. It has the distance function:

d² = x² + y² + z² - ct²

T is an elapsed time. To convert it to space units, multiply it by c. Note that seconds * miles/sec = miles. In fact, one nanosecond of time is almost exactly 12 inches.

4D distance is called the interval between 2 points. If the difference in time is greater than the distance in space, the absolute distance is negative.

The only way around that is to arbitrarily define distance as positive or zero. That's what special relativity scientists do, but then you have to give negative distance another name, like "pseudodistance" or "imaginary space" or "a timelike interval." Verdana♥Bold 08:32, 17 August 2020 (UTC)

Semi-metric space

In the opening sentence, this is also referred to as a semi-metric space. This is not the usual definition of semi-metric space, if there is a usual definition. For example, http://www.ams.org/journals/proc/1961-012-05/S0002-9939-1961-0125562-9/S0002-9939-1961-0125562-9.pdf shows a definition that lines up with Metric_(mathematics)#Semimetrics. I will remove the offending word, because Wikipedia should try to be consistent, and because it seems this is non-standard. 209.194.136.27 (talk) 17:43, 26 July 2016 (UTC)