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I was wondering about a question that I came across in my topology text . The question was to find a metric function in which a ball of smaller radius contains a ball with larger radius . Then I came across the following question :

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

In the last answer , someone has mentioned about non-constant metric function . Did he mean a metric function in which the metric between two points vary depending on let's say context or a metric function which is simply not the same for all the pair of points ? Are there any systems in mathematics which fall under my first interpretation of "non constant metric functions" in the above line ? Can one such function serve as an answer to the question about the bigger ball being inside the smaller ball ?

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itp dusra
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  • It appears that he meant simply a metric that is not the same for all pairs of distinct points. In other words, it’s not simply a multiple of the discrete metric. Since a metric on $X$ is a definite function from $X\times X$ to $\Bbb R$, your first interpretation isn’t possible: such an object wouldn’t be a metric. – Brian M. Scott Sep 04 '16 at 18:45
  • @Brain M scott : Any example of such an object in mathematics ? – itp dusra Sep 05 '16 at 14:54
  • I suppose that one could have a function $f:X\times X\times [0,1]$, say, so that the value $f(x,y,t)$ at a point $\langle x,y\rangle\in X\times X$ would depend not just on $x$ and $y$, but also on a parameter $t\in[0,1]$. If $t$ represents time, and one thinks of the space $X$ as stretching/shrinking, the changing distances between points could be described that way. – Brian M. Scott Sep 05 '16 at 17:26

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