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Can we define a metric $d$ on any set $X$ s.t. $d(x, y)=\infty$ $\forall x, y$, $x\neq y$

In this case, how will X look like? And What will be the open sets of $X$.

Maybe definition allows only real value of $d$. What will be the consequences of defining $d$ in this way?

user772744
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  • A condition of a metric is that its finite. Another is that it is zero if $x=y$. –  Jun 12 '20 at 06:57
  • yeah! So is that any space! where this all is possible? Have anybody ever defined it so? – user772744 Jun 12 '20 at 06:58
  • What you have defined is not a metric. Questions like what are the open sets aren't really valid here. Imagine if the 'distance' between $0$ and $0$ was infinite. What does that even mean? –  Jun 12 '20 at 06:59
  • Have a look at extended metrics – adfriedman Jun 12 '20 at 07:06
  • I think even extended metrics require $d(x,x)=0$. –  Jun 12 '20 at 07:07
  • @tomasliam I should have said for distinct x and y.. then...? – user772744 Jun 12 '20 at 07:10
  • In that case it would kind of be an extension to the discrete metric. As for what properties it has, I am not sure. But I doubt it would be anything meaningfully different. –  Jun 12 '20 at 07:12
  • @tomasliam yeah... thank you. I was just wondering if anybody has come across this kind of space. – user772744 Jun 12 '20 at 07:14
  • Provided that one should have $d(x,y)=0$ if and only if $x=y$ because... well, that obviously makes sense, there is an interesting (perhaps over-enthousiastic) answer by Ittay Weiss here. See also the discussion in the comments with former user hmakholmleftoverMonica over some nuances, mainly the fact that $[0,\infty]$ with the obvious extended metric is homeomorphic to $[0,1)\cup {2}$, and not to $[0,1]$ as the inattentive reader might think. –  Jun 12 '20 at 07:25
  • @Gae.S. Thank you. I will have a look, seems interesting. – user772744 Jun 12 '20 at 07:28

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