Let \(X\) be a metric space . For \(r>0\), the \(r\)-neighbourhood \(S_r\) of a subset \(S\) is given by
\[ S_r=\{x\in X\mid \dist (x,S)< r\}. \]
Remark
- \[ S_r=\cup_{x\in S} B_r(x). \]
- For bounded \(S\), \(x\in S_r^c\) and \(p\in S\), we have \[ d(x,p) \cong d(x,y)\quad \forall y\in S \] (see (0x68d03834) ).