Metrics, open balls, and neighborhoods

A metric on a set X assigns a nonnegative real number to every ordered pair of points. The value is zero exactly when the points agree, reversing the order leaves it unchanged, and traveling directly costs no more than traveling through an intermediate point. That last condition, the triangle inequality, is what allows nearby-to-nearby reasoning. It also prevents a shortcut through a third point from contradicting the claimed distance. These requirements must hold for every pair or triple, rather than just for examples plotted on a screen.

On the real line, absolute difference is a metric, while squared difference is not. For the points 0, 1, and 2, the squared direct distance is four but the two squared legs sum to two. Squared distance remains useful as an optimization loss, but the metric theorems cannot automatically be applied to it. A second warning concerns zero distance: allowing distinct points at distance zero gives a pseudometric. Identifying such points can produce a metric quotient, but that identification is an additional mathematical step.

An open ball centered at x contains points whose distance from x is strictly less than a positive radius. In the Euclidean plane its picture is a disk. With the maximum metric, which takes the larger coordinate difference, its picture is an axis-aligned open square. The word ball denotes the inequality, not a commitment to a round shape. In the discrete metric, every distinct pair has distance one, so a radius at most one isolates its center and a radius greater than one includes the whole space.

Open sets are sets that contain a ball around each of their points. The radius may vary from point to point. To prove that a Euclidean interval is open, a point near its endpoint needs a smaller radius than a point near its center. A neighborhood of x is any set containing an open set around x; it need not itself be open. These distinctions become useful when functions, sequences, or spaces have no useful visual picture but still admit an explicit distance estimate.

Why is an open ball open? Choose a point y inside a ball centered at x with radius r. There is a positive margin between r and d(x,y). If z is closer to y than that margin, the triangle inequality places z inside the original ball. This proof is more valuable than the picture because it works for every metric, including edit distances and distances between functions. The local radius is produced from information already available rather than guessed from a drawing.

Two metrics can have identical open sets even when numerical distances differ. In the plane, Euclidean and maximum distances satisfy inequalities that place a ball for one metric inside a suitable ball for the other. Thus they define the same topology. This equivalence preserves convergence and continuity, but a general agreement of open sets does not by itself preserve Cauchy sequences or completeness. Later lessons will identify why these latter notions require control over pairs throughout a tail of a sequence.