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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?

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    $\begingroup$ 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. $\endgroup$ – user99914 Sep 4 '16 at 21:19
  • $\begingroup$ 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. $\endgroup$ – Daniel Fischer Sep 4 '16 at 21:19
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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).

However when it is not specified it is usually because it is clear that we are working with a specific metric (in your case I would guess the euclidean one).

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