Open Balls and Neighborhoods

Real Analysis · Axiom Academy

LESSON Open Balls and Neighborhoods How a distance function carves out regions — and why the shape of those regions is decided entirely by the metric. Let (X, d) be a metric space. The open ball of radius r about the center x collects every point whose distance from x is strictly less than r . The animation tests every point in the plane against this rule. Points that pass — — turn green; points that fail turn red. The dashed ring is the set of points with d(x,y) = r exactly: it is the boundary, and the strict inequality leaves it out . 2. On the Real Line: an Open Interval Take the most familiar metric of all: with d(x,y) = |x - y| . The open ball about x is the set of reals within r of x on either side — that is the open interval . An open ball on is just an open interval The animation grows the interval out from x . The two endpoints x - r and x + r are drawn as hollow circles: each sits at distance exactly r , so each fails and is excluded — the open-interval brackets made visible. 3. Different Metrics, Different Shapes Here is the surprise. In the same center and the same radius produce wildly different open balls, because the shape of a ball is decided by the metric. Three standard metrics give three iconic shapes.

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