Knowlpedia

Closed balls are closed

Every closed ball in a metric space is a closed subset.

In a (X,d)(X,d), every

B‾(x0,r)={x∈X:d(x,x0)≤r},r≥0,\overline B(x_0,r)=\{x\in X:d(x,x_0)\le r\}, \qquad r\ge0,

is a .

Remarks

If x∉B‾(x0,r)x\notin\overline B(x_0,r), let δ=d(x,x0)−r>0\delta=d(x,x_0)-r>0. If d(y,x)<δd(y,x)<\delta, then

d(y,x0)≥d(x,x0)−d(y,x)>r.d(y,x_0)\ge d(x,x_0)-d(y,x)>r.

Thus B(x,δ)B(x,\delta) lies in the complement, so the complement is open.