Let \(X\) be a topological space. A subset \(U \subseteq X\) is closed in \(X\) if its complement \(X\sm U\) is open in \(X\). If \(Y \subseteq X\) carries the subspace topology and \(U \subseteq Y\), then \(U\) is closed in the subspace \(Y\) if \(Y\sm U\) is open in \(Y\).
Let \(X\) be a topological space and \(U\subseteq X\).
\(U\) is closed in \(X\) if and only if every limit point of \(U\) in \(X\) lies in \(U\).
\(U\) is closed in \(X\) if and only if \(\cl_X(U) = U\), where \(\cl_X(U)\) is the closure of \(U\) in \(X\).
If \(Y\subseteq X\) is a subspace and \(U\subseteq Y\), then \(U\) is closed in \(Y\) if and only if \(U = Y \intersect V\) for some closed subset \(V\) of \(X\).
If \(Y\subseteq X\) is a subspace and \(U\subseteq Y\), then the closure of \(U\) in \(Y\) is \(\cl_Y(U) = \cl_X(U)\intersect Y\).
[@Mun00].