# Does the distance from point to set depend on metrics?

I have come across with a question that asks to prove that, being $X = \left \{ (x,y) \in \mathbb{R^2}; x^2 + y^2 < 1 \right \}$ and $a = (5,0)$, $d(a,X) = 4$.

My initial thought was to choose the euclidean metric and prove it from there. Am I allowed to do so (choosing a specific metric)? Does the distance from a point to a set depend on the metrics chosen?

• Of course that depends on the metric. The definition of $d(\cdot, A)$ involve the metric already. You should ask the people who gave you this question which metric to use. – user99914 Sep 4 '16 at 21:19
• Unless the point belongs to the set, the distance depends on the chosen metric. On $\mathbb{R}^n$ and subsets thereof, unless specified otherwise, one can assume the Euclidean metric. – Daniel Fischer Sep 4 '16 at 21:19

Of course it depends on the metric (it won't be the same if you choose a distance $d$ or a distance $2d$ for instance).