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In an exercise I found that we can write metric topologies by choosing the distance equal to the length of the shortest path connecting two points; however if the space isn't connected then what "additional step is required" to show that its a metric space / topology?

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    Are you talking about a metric on a graph? – Arthur Dec 02 '19 at 18:53
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    If your space is not connected you are probably going to want to renormalize the metric so that it is bounded by some $A$ on each connected component, and then take a value greater than $A$ to represent the distance between points on different components. – Captain Lama Dec 02 '19 at 19:04
  • @arthur Yes, I am.... –  Dec 02 '19 at 19:18
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    As @CaptainLama says, you can normalize the metric so the distance is always less than $1$, so make the distance between points in different connected components $1$. – Rushabh Mehta Dec 02 '19 at 19:30
  • Shouldn't there be endpoints / non adjacent points in disconnected parts with a distance less than one as well? –  Dec 02 '19 at 19:48

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