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Open Ball — Definition, Formula & Examples

An open ball is the set of all points whose distance from a given center point is strictly less than a specified radius. It generalizes the idea of an open interval on the real line to higher dimensions and abstract metric spaces.

Given a metric space (X,d)(X, d), a point x0∈Xx_0 \in X, and a real number r>0r > 0, the open ball of radius rr centered at x0x_0 is the set B(x0,r)={x∈X:d(x,x0)<r}B(x_0, r) = \{x \in X : d(x, x_0) < r\}.

Key Formula

B(x0,r)={x∈X:d(x,x0)<r}B(x_0, r) = \{x \in X : d(x, x_0) < r\}
Where:
  • x0x_0 = The center point of the ball
  • rr = The radius (a positive real number)
  • d(x,x0)d(x, x_0) = The distance from point x to the center, measured using the metric d
  • XX = The underlying metric space

How It Works

To determine whether a point xx belongs to the open ball B(x0,r)B(x_0, r), compute the distance d(x,x0)d(x, x_0) using whatever metric applies. If that distance is strictly less than rr, the point is inside the ball. Open balls serve as the basic building blocks for defining open sets: a set UU is open if every point in UU has some open ball around it that fits entirely inside UU. This in turn provides the foundation for defining limits, continuity, and convergence in metric spaces.

Worked Example

Problem: In R2\mathbb{R}^2 with the standard Euclidean metric, determine whether the point (3,4)(3, 4) belongs to the open ball B((0,0),6)B((0,0), 6).
Step 1: Compute the Euclidean distance from (3,4)(3, 4) to the center (0,0)(0, 0).
d((3,4), (0,0))=32+42=9+16=25=5d\bigl((3,4),\,(0,0)\bigr) = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
Step 2: Compare this distance to the radius r=6r = 6.
5<65 < 6
Step 3: Since the distance is strictly less than the radius, the point lies inside the open ball.
Answer: Yes, (3,4)∈B((0,0),6)(3, 4) \in B((0,0), 6) because its distance from the origin is 55, which is less than 66.

Why It Matters

Open balls provide the ϵ\epsilon-neighborhoods used in the rigorous ϵ\epsilon-δ\delta definitions of limits and continuity. In a real analysis or topology course, nearly every proof about convergence, compactness, or connectedness relies on open balls or the open sets they generate.

Common Mistakes

Mistake: Including points at exactly the boundary distance, i.e., treating d(x,x0)≤rd(x, x_0) \leq r as the condition.
Correction: An open ball uses the strict inequality d(x,x0)<rd(x, x_0) < r. Including the boundary gives you a closed ball, denoted B‾(x0,r)\overline{B}(x_0, r).