Let \(X\) be a topological space and \(U\subseteq X\). The closure of \(U\) in \(X\) is the intersection of all closed subsets of \(X\) containing \(U\), \[ \cl_X(U) \coloneqq \Intersect_{\substack{B\supseteq U \\ B \text{ closed in } X}} B ,\] also written \(\bar U\).
If \(Y\subseteq X\) is a subspace containing \(U\), the closure of \(U\) in \(Y\) is \(\cl_Y(U) = \cl_X(U) \intersect Y\) [@Mun00]. A point \(x\in X\) lies in \(\cl_X(U)\) if and only if every neighborhood of \(x\) in \(X\) intersects \(U\) [@Mun00].