Let \((X,d)\) be a metric space, let \(p\in X\), and let \(E\subseteq X\).
For \(r>0\), the neighborhood of \(p\) of radius \(r\) is \(N_r(p)\coloneqq\ts{q\in X\st d(p,q)<r}\).
The point \(p\) is a limit point of \(E\) if every neighborhood of \(p\) contains a point \(q\in E\) with \(q\neq p\).
If \(p\in E\) and \(p\) is not a limit point of \(E\), then \(p\) is an isolated point of \(E\).
\(E\) is closed if every limit point of \(E\) belongs to \(E\).
The point \(p\) is an interior point of \(E\) if some neighborhood \(N\) of \(p\) satisfies \(N\subseteq E\).
\(E\) is open if every point of \(E\) is an interior point of \(E\).
The complement of \(E\) is \(E^c\coloneqq\ts{p\in X\st p\notin E}\).
\(E\) is perfect if \(E\) is closed and every point of \(E\) is a limit point of \(E\).
\(E\) is bounded if there are \(M\in\RR\) and \(q\in X\) with \(d(p,q)<M\) for all \(p\in E\).
\(E\) is dense in \(X\) if every point of \(X\) is a limit point of \(E\) or a point of \(E\), or both.
A point \(p\in X\) is a limit point of \(E\) if and only if \((N_r(p)\setminus\ts{p})\cap E\neq\emptyset\) for every \(r>0\). By (c), every point of \(E\) is either a limit point or an isolated point of \(E\), and not both.