deepdream of
          a sidewalk
Show Question
Math and science::Topology

Metric space. ε-balls

Let X be a metric space, let xX and let ε>0 be a real. The open ε-ball around x (or in more detail, the open ball around x of radius ε) is the subset of X given by

B(x,ε)={yX:d(x,y)<ε}

Similarly, the closed ε-ball around x is

B¯(x,ε)={yX:d(x,y)ε}

The definition of open sets in metric spaces is formulated in terms of open ε-balls.

Example

Metrics such as d1, d2 and d for both Rn and continuous function spaces C[a,b], and other metrics such as Hamming distance, each induce ε-balls within the space they are applied in. It is interesting to consider what the ε-ball in each metric space is.


Context