When the Discrete Topology Does Not Imply the Discrete Metric

Metric Space

You may be familiar with the discrete metric on an arbitrary set . The discrete metric makes every subset of an open set. If we let denote the topology induced by the discrete metric on , then

Now, is the converse true?

Recall that the discrete metric function on is defined as:

So, we can say that every Cauchy sequence in is convergent in .

Suppose we have a metric on such that the topology induced by is

Does this imply that is the discrete metric? No.


Counterexample

Define a metric as:

where denotes the usual Euclidean metric on .

  • Every subset of is open in this metric space. Thus,

  • However, this space is not complete. Consider the sequence . It is Cauchy but not convergent in .

This counterexample shows that even if the topology induced by a metric is discrete topology, the metric itself need not be the discrete metric.

Moreover, if you are familiar with the concept of equivalent metrics, observe that is homeomorphic to the metric space . Hence, this example illustrates that completeness is not preserved under homeomorphism.

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