Metric spaces · Definition

Open ball

topologymetric spaces

Before you start you need to know

The open ball is the basic unit of «near a point»: it defines open sets and, from them, limits and continuity.

Definition

The open ball \(B(x,r)\) with center \(x\) and radius \(r>0\) in a metric space \((X,d)\) is the set of points of \(X\) whose distance to \(x\) is less than \(r\).

\[B(x,r)=\{\,y\in X : d(x,y)

In the usual \(\mathbb{R}\), \(B(x,r)\) is the interval \((x-r,x+r)\); in Euclidean \(\mathbb{R}^2\), a disc without its circle. The inequality is strict, so points at distance exactly \(r\) are outside. And always \(x\in B(x,r)\), because \(d(x,x)=0<r\): no ball is empty.

The shape of the ball depends on the metric. With the max distance, \(B((0,0),1)\) is the square \((-1,1)^2\); with the taxicab one, a rhombus of area \(2\); with the discrete one, \(B(x,1)=\{x\}\) and \(B(x,r)=X\) if \(r>1\). So always keep in mind which \(d\) is in use.

A property used constantly: if \(y\in B(x,r)\) and \(s=r-d(x,y)>0\), then \(B(y,s)\subseteq B(x,r)\). It follows from the triangle inequality: if \(d(y,z)<s\), then \(d(x,z)\le d(x,y)+d(y,z)<r\). In other words, from any point of a ball you can take a small step without leaving it.

Two balls may intersect, but the intersection of two balls contains a ball around each of its points, thanks to the previous property. This is exactly what is needed to turn balls into the base of a topology.

Think about it

In the usual \(\mathbb{R}\), when is \(B(a,r)\subseteq B(b,s)\)?

When \((a-r,a+r)\subseteq(b-s,b+s)\), that is, \(|a-b|+r\le s\).

Can a ball with a large radius have a single point?

Yes, if the radius is small relative to the distances in the space: in the discrete metric, \(B(x,1)=\{x\}\). With radius larger than 1 it would be the whole space.

Exercises 10

  1. In \(\mathbb{R}\) with \(d(x,y)=|x-y|\), which interval is the open ball \(B(2,3)\)?
  2. In the usual \(\mathbb{R}\), which of these points belongs to \(B(0,2)\)?
  3. In Euclidean \(\mathbb{R}^2\), which point lies in \(B((0,0),1)\)?
  4. With the max distance \(d_\infty\) on \(\mathbb{R}^2\), the ball \(B((0,0),2)\) is a square. What is its area?
  5. With the taxicab distance \(d_1\) on \(\mathbb{R}^2\), the ball \(B((0,0),1)\) is a rhombus with vertices \((\pm1,0)\) and \((0,\pm1)\). What is its area?
  6. In a space with the discrete metric, what is the ball \(B(a,1)\)?
  7. In the usual \(\mathbb{R}\), what is the intersection \(B(0,2)\cap B(3,2)\)?
  8. How many points \((a,b)\) with integer coordinates lie in the Euclidean ball \(B((0,0),3)\)?
  9. Let \(y\in B(a,r)\) and \(s=r-d(a,y)\). In \(\mathbb{R}\) with \(a=0\), \(r=5\) and \(y=3\), what is \(s\), the radius ensuring \(B(y,s)\subseteq B(a,r)\)?
  10. Which statement holds for every open ball \(B(x,r)\) with \(r>0\) in any metric space?

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §20
  • Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 2