Consider finding a a point with rational distance to the corners of unit square.
Under the Euclidean metric this is very hard. (unsolved)
Under the "city block" or taxicab metric this is very easy namely any rational point.
Yet both metrics give rise to the exact same topology. Is there a way to transform a point form one metric to the other that preserves rationality? What gives rise to this discrepancy?
The topology doesn't care about exact distance values in the metric; it only cares about what points are close to what others in a broad sense. So for a question where the exact distance values are important, you wouldn't expect the question to say the same under different metrics even if they induce the same topology.