Let \((X,d)\) be a metric space, and \(c\in X\) a point, and \(r > 0\). The set \[B(c,r):=\{x\in X:d(c,x) < r \}\] is called the open ball with center \(c\) and radius \(r\) with respect to the metric \(d\).
Given a point \(c\in X\), any subset \(N\subset X\) with the following properties:
(1) \(c\in N\) and
(2) There exists an \(\epsilon > 0\) with \(B(c,\epsilon)\subset N\)
is called the neighborhood of \(c\). In particular, the open ball \(B(c,\epsilon)\) itself is a neighborhood of \(c\). It is called the \(\epsilon\)-neighborhood of \(c\).
Definitions: 1 2 3 4 5 6 7
Parts: 8 9
Proofs: 10 11 12 13 14 15 16
Propositions: 17