Q:

uniformly discrete in one metric but not in the other giving the same topology

Is it possible to define two metrics on $X$ giving the same topologies such that there exists a nonempty subset of $X$ which is uniformly discrete in one but not in the other?

A:

Yes, the sequence $\{2^{-n}:n\in\omega\}$ is not uniformly discrete with respect to the metric it inherits from the real line, but it is uniformly discrete with respect to the discrete metric.

