0.1 Epsilon Ball
In metric spaces we need to establish regions around points (regions) of interest so that we may constrict or expand them as we wish, We do so by defining a general concept of a ball of all points that are at most a certain “distance” away from the point of interest.
On [[knowledge-mathematics-analysis-functional-analysis-metric-spaces|metric space]] \((X,d)\) an open epsilon ball \(B_{\epsilon}(x)\) is a generalized ball of radius \(\epsilon\) and centre \(x\) that does not contain its boundary points. This is described mathematically as \[ B_\epsilon(x):=\{y\in X:d(x,y)<\epsilon\} \] Similarly, a closed epsilon ball \(\bar B_\epsilon(x)\) is the same but this time does contain its boundary points, given by \[ \bar B_{\epsilon}(x):=\{y\in X:d(x,y)\leq \epsilon\} \]