Let \((X,d)\) be a metric space, let \(A\subseteq X\), and let \(p\in X\).
A neighborhood of \(p\) is an open set \(U_p\subseteq X\) containing \(p\).
An \(\varepsilon\)-neighborhood of \(p\) is an open ball \(B_\varepsilon(p) \coloneqq \theset{q\in X \suchthat d(p, q) < \varepsilon}\) for some \(\varepsilon>0\).
The point \(p\) is an accumulation point or a limit point of \(A\) if every punctured neighborhood \(U_p\setminus\theset{p}\) contains a point \(q\in A\). Write \(A'\) for the set of limit points of \(A\) and \(\overline{A} \coloneqq A\cup A'\) for the closure of \(A\).
If \(p\in A\) and \(p\) is not a limit point of \(A\), then \(p\) is an isolated point of \(A\).
\(A\) is closed if \(A' \subseteq A\), so that \(A\) contains all of its limit points.
A point \(p\in A\) is an interior point of \(A\) if there is a neighborhood \(U_p\) of \(p\) with \(U_p \subseteq A\).
\(A\) is open if every point of \(A\) is an interior point of \(A\).
\(A\) is perfect if \(A\) is closed and \(A\subseteq A'\), so that every point of \(A\) is a limit point of \(A\).
\(A\) is bounded if there are a real number \(M\) and a point \(q\in X\) such that \(d(p, q) < M\) for all \(p\in A\).
\(A\) is dense in \(X\) if every point of \(X\) lies in \(A\) or is a limit point of \(A\), that is, \(X = A\cup A' = \overline{A}\).