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from the actual definition of metric space ,we know that

metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality i am interested what is a symmetric distance?i know triangle equality,something sum of two length is more then third one,but what about symmetric distance?thanks in advance

dato datuashvili
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2 Answers2

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Saying that the metric (or distance) is symmetric just means that the distance from $x$ to $y$ is always the same as the distance from $y$ to $x$. In symbols, for all $x,y\in X$ we have $$d(x,y)=d(y,x)\;.$$

Brian M. Scott
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  • thanks in advance,thanks guys – dato datuashvili Sep 18 '13 at 07:02
  • could there be non symmetric distance? – dato datuashvili Sep 18 '13 at 07:03
  • @dato: People have looked at non-symmetric distance functions, because they have real-world applications. In terms of effort, for instance, the ‘distance’ up a hill is longer than the ‘distance’ down that hill if you’re a runner or cyclist. They’re a lot harder to work with, however, and they don’t have nearly so nice an associated body of theory. – Brian M. Scott Sep 18 '13 at 07:05
  • but in terms of algebraic does there exist non symmetric distance? – dato datuashvili Sep 18 '13 at 07:06
  • @dato: One can write down asymmetric functions that model the kind of psychological ‘distance’ that I mentioned in the other comment, but they aren’t usually thought of as distance functions; in my (limited) experience they’re more likely to be thought of as measuring energy expenditure, work done, or some psychological analogue of those. – Brian M. Scott Sep 18 '13 at 07:09
  • non-symmetric metric spaces go by the name of quasimetric spaces and they are extremely important in many areas including computer science. Their theory is quite a bit different to the theory of symmetric metric spaces. Also, see Finslerian geometry in non-Riemannian geometry - very important concept in physics. – Ittay Weiss Sep 18 '13 at 07:15
  • And indeed @BrianM.Scott is very right in saying the body of theory is not nearly as nice as in the symmetric case. The non-symmetry causes lots of difficulties. For instance, even the notion of completion bifurcates as there are many natural but non-equivalent notions of Cauchy sequences. – Ittay Weiss Sep 18 '13 at 07:19
  • @Ittay: Yes, I knew the term quasimetric space; I just didn’t want to get too technical, since I know that all of this is very new to the OP. I don’t know much of anything about their applications, however. – Brian M. Scott Sep 18 '13 at 07:19
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That is, we can't have any $x,y$ in the set such that $d(x,y)>d(y,x)$. We must have $d(x,y)=d(y,x)$ for all $x,y$ in the set. (We want the distance from the one to the other to be the same as the distance from the other to the one, since that's how distance actually works "in real life.")

Cameron Buie
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