n. 1. (between two points in Euclidean space) the length of the shortest line segment joining the given points, measured as the square root of the sums of the squares of the differences between the coordinates of the two points; if A and B are respectively the points (a₁, a₂) and (b₁, b₂) in the Cartesian plane, then the length since, as shown in the figure belowAB is the hypotenuse of a right-angled triangle whose other sides are the differences of the coordinates of A and B. In n-dimensional Euclidean space, 2. any length measured along a curve, especially straight line or great circle. See also geodesic. 3. (from a point to a line) see perpendicular distance.
4a. (in a topological space) a metric. See also norm.
b. (between sets in a metric space) the infimum of the distance between points in one set and points in the other.