I understand that the use of defining a topology on a set is to define what it means for a map out of the space to be continuous. It is then natural to ask for which types of maps out of a set is it possible to define a topology on the set to allow only said maps to be continuous (e.g. any map continuity -> discrete topology, 'traditional' continuity -> euclidean topology).

This has not yet been addressed in the textbook I am currently reading so I decided to take it here.

My question is specifically about isometric maps, but any more general answer will suffice.


1 Answer 1


In order to talk about "isometry", you need a "metric".

In a general topological space, this is no metric and the topology is not necessarily metrizable.

If you do have a metric space $(X,d)$, then the natural topology is the metric topology induced by the metric $d$. One can then ask whether a continuous function with respect to this topology is an isometry.

But there is no reason for an arbitrary continuous map to be an isometry: you can always consider the constant map.

  • $\begingroup$ Ok, but obviously the natural topology on such a metric space doesn't cut it. But is it possible to define some other topology on the metric space that does the job is my question. $\endgroup$ Jan 14, 2021 at 13:21
  • $\begingroup$ @KillianHeanue: whatever topology you define, you can always consider the constant map, which is continuous. $\endgroup$
    – user9464
    Jan 14, 2021 at 13:22
  • $\begingroup$ that's not an issue as the constant map is an isometry. $\endgroup$ Jan 14, 2021 at 13:23
  • $\begingroup$ @KillianHeanue: no, the constant map is not an isometry. The constant map is not the same as the "identity" map. $\endgroup$
    – user9464
    Jan 14, 2021 at 13:24
  • 1
    $\begingroup$ @KillianHeanue: sorry I have to go now. you can leave questions there and I may be back. $\endgroup$
    – user9464
    Jan 14, 2021 at 13:54

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